*Article* **Local Antimagic Chromatic Number for Copies of Graphs**

**Martin Baˇca 1,†, Andrea Semaniˇcová-Fe ˇnovˇcíková 1,\*,† and Tao-Ming Wang 2,†**


**Abstract:** An edge labeling of a graph *G* = (*V*, *E*) using every label from the set {1, 2, ... , |*E*(*G*)|} exactly once is a *local antimagic labeling* if the vertex-weights are distinct for every pair of neighboring vertices, where a vertex-weight is the sum of labels of all edges incident with that vertex. Any *local antimagic labeling* induces a proper vertex coloring of *G* where the color of a vertex is its vertex-weight. This naturally leads to the concept of a local antimagic chromatic number. The local antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of *G* induced by *local antimagic labelings* of *G*. In this paper, we estimate the bounds of the local antimagic chromatic number for disjoint union of multiple copies of a graph.

**Keywords:** *local antimagic labeling*; local antimagic chromatic number; copies of graphs

**MSC:** 05C78; 05C69

#### **1. Introduction**

In this paper, we will consider only finite graphs without loops or multiple edges. For graph theoretic terminology we refer to the book by Chartrand and Lesniak [1].

An antimagic labeling of a graph *G* = (*V*, *E*) is a bijection *f* from the set of edges of *G* to the integers {1, 2, ... , |*E*(*G*)|} such that all vertex-weights are pairwise distinct, where a vertex-weight is the sum of labels of all edges incident with that vertex, i.e., for the vertex *<sup>u</sup>* <sup>∈</sup> *<sup>V</sup>*(*G*) the weight *wt*(*u*) = <sup>∑</sup> *uv*∈*E*(*G*) *f*(*uv*). A graph is called *antimagic* if it admits an

antimagic labeling.

The concept of *antimagic labeling* was introduced by Hartsfield and Ringel [2] who conjectured that every simple connected graph, other than *K*2, is antimagic. This conjecture is still open although for some special classes of graphs it was proved, see for instance [3–8]. Alon et al. [9] proved that large dense graphs are antimagic. Hefetz et al. [10] proved that any graph on *p<sup>k</sup>* vertices that admits a *Cp*-factor, where *p* is an odd prime and *k* is a positive integer, is antimagic. Perhaps the most remarkable result to date is the proof for regular graphs of odd degree given by Cranston et al. in [11], which was subsequently adapted to regular graphs of even degree by Bércz et al. in [12] and by Chang et al. in [13].

Recently, two groups of authors in [14,15] independently introduced a *local antimagic labeling* as local version of the Hartsfield and Ringel's concept of antimagic labeling. An edge labeling using every label from the set {1, 2, ... , |*E*(*G*)|} exactly once is a *local antimagic labeling* if the vertex-weights *wt*(*u*) and *wt*(*v*) are distinct for every pair of neighboring vertices *u*, *v*.

In [14] authors conjectured that any connected graph other than *K*<sup>2</sup> admits a *local antimagic labeling*. Bensmail et al. [15] propose the slightly stronger form of the previous conjecture that every graph without component isomorphic to *K*<sup>2</sup> has a *local antimagic labeling*. This conjecture was proved by Haslegrave [16] using the probabilistic method.

**Citation:** Baˇca, M.; Semaniˇcová-Fe ˇnovˇcíková, A.; Wang, T.-M. Local Antimagic Chromatic Number for Copies of Graphs. *Mathematics* **2021**, *9*, 1230. https://doi.org/10.3390/ math9111230

Academic Editors: Janez Žerovnik and Darja Rupnik Poklukar

Received: 1 May 2021 Accepted: 26 May 2021 Published: 27 May 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Any *local antimagic labeling* induces a proper vertex coloring of *G* where the vertexweight *wt*(*u*) is the color of *u*. This naturally leads to the concept of a local antimagic chromatic number introduced in [14]. The *local antimagic chromatic number χla*(*G*) is defined to be the minimum number of colors taken over all colorings of *G* induced by *local antimagic labelings* of *G*.

For any graph *G*, *χla*(*G*) ≥ *χ*(*G*), where *χ*(*G*) is the chromatic number of *G* as the minimum number of colors needed to produce a proper coloring of *G*. In [14] is investigated the local antimagic chromatic number for paths, cycles, friendship graphs, wheels and complete bipartite graphs. Moreover, there is proved that for any tree *T* with *l* leaves *χla*(*T*) ≥ *l* + 1.

In this paper, we investigate the local antimagic chromatic number for disjoint union of multiple copies of a graph *G*, denoted by *mG*, *m* ≥ 1, and we present some estimations of this parameter.

Please note that *G* does not have to be necessarily connected. By the symbol *xi* we denote the element (vertex or edge) corresponding to the element (vertex or edge) *x* in the *i*th copy of *G* in *mG*, *i* = 1, 2, . . . , *m*.

#### **2. Graphs with Vertices of Even Degrees**

A graph *G* is called *equally* 2*-edge colorable* if it is possible to color its edges with two colors *c*1, *c*<sup>2</sup> such that for every vertex *v* ∈ *V*(*G*) the number of edges incident to the vertex *v* colored with color *c*<sup>1</sup> is the same as the number of edges incident to the vertex *v* colored with color *<sup>c</sup>*2. This means that for any vertex *<sup>v</sup>* <sup>∈</sup> *<sup>V</sup>*(*G*) is *<sup>n</sup>*1(*v*) = *<sup>n</sup>*2(*v*), where *<sup>n</sup><sup>i</sup>* (*v*) denotes the number of edges incident to the vertex *v* and colored with color *ci*, *i* = 1, 2. Trivially, if a graph *G* is equally 2-edge colorable then all vertices in *G* have even degrees.

Consider that *G* is an even regular graph. Then there exists an Euler circle in *G*. If we alternatively color the edges in the Euler circle with colors *c*<sup>1</sup> and *c*<sup>2</sup> we obtain that either for every vertex *v* in *G* holds *n*1(*v*) = *n*2(*v*), or there exists exactly one vertex in *G*, say *w*, such that *n*1(*w*) = *n*2(*w*) + 2.

Consider a 2-edge coloring *c* of a graph *G*. Let *c*(*G*) denote the number of vertices in *G* such that *n*<sup>1</sup> *<sup>c</sup>* (*v*) <sup>=</sup> *<sup>n</sup>*<sup>2</sup> *<sup>c</sup>* (*v*) under the labeling *c*. In this case we say that *c* is a *c*(*G*)*-equally* 2*-edge coloring* of *G*.

Let *c* be any 2-edge coloring of *G*. Let *f* be any bijective mapping in *G*, *f* : *E*(*G*) → {1, 2, . . . , |*E*(*G*)|}. We define an edge labeling *g* of *mG*, *m* ≥ 1 in the following way

$$g(e\_i) = \begin{cases} m(f(e) - 1) + i, & \text{if } c(e) = c\_1 \text{ and } i = 1, 2, \dots, m, \\ mf(e) + 1 - i, & \text{if } c(e) = c\_2 \text{ and } i = 1, 2, \dots, m. \end{cases}$$

If an edge in *G* is labeled with the number *t*, 1 ≤ *t* ≤ |*E*(*G*)|, then the corresponding edges in *mG* are labeled with numbers from the set {*m*(*t* − 1) + 1, *m*(*t* − 1) + 2, ... , *mt*}. Thus we immediately obtain that the labeling *g* is a bijective mapping that assigns numbers 1, 2, . . . , *m*|*E*(*G*)| to the edges of *mG*.

Moreover, for the weight of the vertex *vi*, *i* = 1, 2, ... , *m*, in *mG* under the labeling *g* we obtain the following

$$\begin{split} wt\_{\mathcal{S}}(v\_{i}) &= \sum\_{uv \in E(G)} g(u\_{i}v\_{i}) = \sum\_{uv \in E(G) \colon c(uv) = c\_{1}} g(u\_{i}v\_{i}) + \sum\_{uv \in E(G) \colon c(uv) = c\_{2}} g(u\_{i}v\_{i}) \\ &= \sum\_{uv \in E(G) \colon c(uv) = c\_{1}} \left( m(f(uv) - 1) + i \right) + \sum\_{uv \in E(G) \colon c(uv) = c\_{2}} \left( mf(uv) + 1 - i \right) \\ &= m \sum\_{uv \in E(G) \colon c(uv) = c\_{1}} f(uv) + (i - m)n\_{c}^{1}(v) \\ &+ m \sum\_{uv \in E(G) \colon c(uv) = c\_{2}} f(uv) + (1 - i)n\_{c}^{2}(v) \\ &= m \sum\_{uv \in E(G)} f(uv) + (i - m)n\_{c}^{1}(v) + (1 - i)n\_{c}^{2}(v) \end{split}$$

$$=\boldsymbol{m}\cdot\boldsymbol{w}t\_f(\boldsymbol{v}) + (\boldsymbol{i}-\boldsymbol{m})\boldsymbol{n}\_c^1(\boldsymbol{v}) + (1-\boldsymbol{i})\boldsymbol{n}\_c^2(\boldsymbol{v}).$$

Thus, for every vertex *<sup>v</sup>* <sup>∈</sup> *<sup>V</sup>*(*G*) such that *<sup>n</sup>*<sup>1</sup> *<sup>c</sup>* (*v*) = *n*<sup>2</sup> *<sup>c</sup>* (*v*) = deg(*v*)/2 we obtain

$$wt\_{\mathcal{S}}(v\_i) = m \cdot wt\_f(v) + \frac{(1 - m)\deg(v)}{2} \tag{1}$$

for *i* = 1, 2, ... , *m*. This means that the corresponding vertices in different copies have the same weights. Summarizing the previous we obtain the following lemma that will be used later.

**Lemma 1.** *Let G be a graph and let c be a* 2*-edge coloring of G and let f , f* : *E*(*G*) → {1, 2, ... , |*E*(*G*)|}*, be a bijection. Let m, m* ≥ 1*, be a positive integer. Then there exists an edge labeling g of mG such that the weights of vertices vi, i* = 1, 2, ... , *m, corresponding to the vertex <sup>v</sup>* <sup>∈</sup> *<sup>V</sup>*(*G*) *satisfying n*<sup>1</sup> *<sup>c</sup>* (*v*) = *n*<sup>2</sup> *<sup>c</sup>* (*v*) = deg(*v*)/2 *will be the same.*

Immediately from the previous result we obtain the following theorem for equally 2-edge colorable graphs.

**Theorem 1.** *Let m be a positive integer. Let G be an equally* 2*-edge colorable graph and let f be a local vertex antimagic edge labeling of G that uses χla*(*G*) *colors. Let for every edge uv* ∈ *E*(*G*) *be*

$$mwt\_f(v) + \frac{(1-m)\deg(v)}{2} \neq mwt\_f(u) + \frac{(1-m)\deg(u)}{2}.$$

*Then*

$$
\chi\_{la}(m\mathcal{G}) \le \chi\_{la}(\mathcal{G}).
$$

**Proof.** Let *f* be a local vertex antimagic edge labeling of *G* that uses *χla*(*G*) colors. Let *c* be an equally 2-edge coloring of *<sup>G</sup>*. This means that for every vertex *<sup>v</sup>* <sup>∈</sup> *<sup>V</sup>*(*G*) is *<sup>n</sup>*1(*v*) = *n*2(*v*) = deg(*v*)/2.

According to Lemma 1 and Equality (1) we obtain that there exists a labeling *g* of *mG*, *m* ≥ 1, such that for every *v* ∈ *V*(*G*) and every *i* = 1, 2, ... , *m* holds *wtg*(*vi*) = *m* · *wtf*(*v*) + (1 − *m*) deg(*v*)/2. Thus, *g* is such labeling that the corresponding vertices in different copies have the same weights. If for all adjacent vertices *u*, *v* ∈ *V*(*G*) holds

$$mwt\_f(v) + \frac{(1-m)\deg(v)}{2} \neq mwt\_f(u) + \frac{(1-m)\deg(u)}{2} \tag{2}$$

then also all adjacent vertices in *mG* have distinct weights. Moreover, *χla*(*mG*) ≤ *χla*(*G*). This concludes the proof.

Note, if *G* is a regular graph then the condition (2) trivially holds. Results in the next two theorems are based on the Petersen Theorem.

**Proposition 1. (Petersen Theorem)** *Let G be a* 2*r-regular graph. Then there exists a* 2*-factor in G.*

Notice that after removing edges of the 2-factor guaranteed by Petersen Theorem we have again an even regular graph. Thus, by induction, an even regular graph has a 2-factorization.

**Theorem 2.** *Let G be a* 4*r-regular graph, r* ≥ 1*. Then for every positive integer m*

$$
\chi\_{la}(m\mathcal{G}) \le \chi\_{la}(\mathcal{G}).
$$

**Proof.** Let *G* be a 4*r*-regular graph. According to Petersen Theorem *G* is decomposable into 2-factors *F*1, *F*2,..., *F*2*r*. Consider an edge coloring *c* of *G* defined such that

$$\mathcal{c}(e) = \begin{cases} c\_{1\prime} & \text{if } e \in E(F\_{\vec{j}}), \, \vec{j} = 1, 2, \dots, r, \\ c\_{2\prime} & \text{if } e \in E(F\_{\vec{j}}), \, \vec{j} = r + 1, r + 2, \dots, 2r. \end{cases}$$

Evidently, *c* is an equally 2-edge coloring of *G*. Thus, immediately according to Theorem 1 we obtain the desired result.

**Theorem 3.** *Let G be a* (4*r* + 2)*-regular graph, r* ≥ 0*, containing a* 2*-factor consisting only from even cycles. Then for every positive integer m*

$$
\chi\_{la}(m\mathbf{G}) \le \chi\_{la}(\mathbf{G}).
$$

**Proof.** Let *G* be a (4*r* + 2)-regular graph containing a 2-factor consisting only from even cycles. Denote this 2-factor by *F*1. Let us denote the edges in component *F*<sup>1</sup> by the symbols *e*1,*e*2, ... ,*e*|*VG*<sup>|</sup> arbitrarily in such a way that all cycles in *F*<sup>1</sup> are of the form *eses*+1*es*+<sup>2</sup> ...*es*+*t*, where *s*, *t* are odd integers. As all cycles in *F*<sup>1</sup> are even, evidently every vertex in *G* is incident with an edge in *F*<sup>1</sup> with an even and also with an odd index.

According to Petersen Theorem the graph *G* − *F*<sup>1</sup> is decomposable into 2-factors *F*2, *F*3, ... , *F*2*r*+1. Consider an edge coloring *c* of *G* defined such that

$$c(e) = \begin{cases} c\_{1\prime} & \text{if } e \in E(F\_1), \, e = e\_{2i-1\prime}, i = 1, 2, \dots, \frac{|V(G)|}{2}, \\ & \text{or if } e \in E(F\_{\overline{\}}), \, j = 2, 3, \dots, r + 1, \\ c\_{2\prime} & \text{if } e \in E(F\_1), \, e = e\_{2i}, \, i = 1, 2, \dots, \frac{|V(G)|}{2}, \\ & \text{or if } e \in E(F\_{\overline{\}}), \, j = r + 2, r + 3, \dots, 2r + 1. \end{cases}$$

It is easy to see that for every vertex *v* ∈ *V*(*G*) holds

$$n^1(v) = n^2(v) = 2r + 1.$$

This means that *c* is an equally 2-edge coloring of *G*. By Theorem 1 we obtain that *χla*(*mG*) ≤ *χla*(*G*).

**Corollary 1.** *Let n, m be positive integers, n* ≥ 2*, m* ≥ 1*. Then*

$$
\chi\_{la}(m\mathcal{C}\_{2n}) = \mathfrak{3}.
$$

**Proof.** In [14] it was proved that *χla*(*Ck*) = 3 for every *k* ≥ 3. According to Theorem 3 we obtain that if *k* = 2*n* then for every positive integer *m* holds *χla*(*mC*2*n*) ≤ 3.

Now suppose there exists a *local antimagic labeling f* that induces a 2-coloring C of *mC*2*n*, i.e., the set of the vertex weights consists of two numbers C<sup>1</sup> and C2. As every edge label contributes exactly once to the vertex weight of a vertex colored C<sup>1</sup> we obtain

$$mm \cdot \mathcal{C}\_1 = 1 + 2 + \dots + 2nm.$$

However, every edge label contributes also exactly once to the vertex weight of a vertex colored C<sup>2</sup> thus

$$mm \cdot \mathcal{C}\_2 = 1 + 2 + \dots + 2nm.$$

A contradiction. Thus, *χla*(*mC*2*n*) ≥ 3.

**Theorem 4.** *Let n, m be positive integers, n* ≥ 1*, m* ≥ 1*. Then*

$$
\chi\_{la}(m\mathcal{C}\_{2n+1}) \le m+2.
$$

**Proof.** Let us denote the vertex set and the edge set of *mC*2*n*+<sup>1</sup> such that *V*(*C*2*n*+1) = {*v j <sup>i</sup>* : *i* = 1, 2, ... , 2*n* + 1, *j* = 1, 2, ... , *m*} and *E*(*C*2*n*+1) = {*v j i v j <sup>i</sup>*+<sup>1</sup> : *i* = 1, 2, ... , 2*n*, *j* = 1, 2, ... , *m*}∪{*v j* 1*v j* <sup>2</sup>*n*+<sup>1</sup> : *j* = 1, 2, ... , *m*}. Let *e j <sup>i</sup>* = *v j i v j <sup>i</sup>*+1, *i* = 1, 2, ... , 2*n*, *j* = 1, 2, ... , *m* and let *e j* <sup>2</sup>*n*+<sup>1</sup> = *v j* 1*v j* <sup>2</sup>*n*+1, *j* = 1, 2, . . . , *m*.

We define an edge labeling *f* of *mC*2*n*+<sup>1</sup> in the following way

$$f(e\_i^j) = \begin{cases} \frac{m(i-1)}{2} + j, & \text{if } i = 1, 3, \dots, 2n + 1, j = 1, 2, \dots, m, \\ m\left(2n + 2 - \frac{i}{2}\right) + 1 - j, & \text{if } i = 2, 4, \dots, 2n, j = 1, 2, \dots, m. \end{cases}$$

For the weight of the vertex *v j i* , *i* = 3, 5, . . . , 2*n* + 1, *j* = 1, 2, . . . , *m* we obtain

$$wt\_f(v\_i^j) = f(e\_{i-1}^j) + f(e\_i^j) = \left[ m\left(2n + 2 - \frac{i-1}{2}\right) + 1 - j \right] + \left[ \frac{m(i-1)}{2} + j \right] = m(2n+2) + 1$$

and for *i* = 2, 4, . . . , 2*n*, *j* = 1, 2, . . . , *m*, we obtain

$$\begin{aligned} wt\_f(v\_i^j) &= f(e\_{i-1}^j) + f(e\_i^j) = \left[ \frac{m((i-1)-1)}{2} + j \right] + \left[ m\left(2n+2-\frac{i}{2}\right) + 1 - j \right] \\ &= m(2n+1) + 1. \end{aligned}$$

The weight of the vertex *v j* <sup>1</sup>, *j* = 1, 2, . . . , *m*, is

$$wt\_f(v\_1^j) = f(e\_1^j) + f(e\_{2n+1}^j) = \left[\frac{m(1-1)}{2} + j\right] + \left[\frac{m((2n+1)-1)}{2} + j\right] = mn + 2j.$$

thus the weights are *mn* + 2, *mn* + 4, ... , *m*(*n* + 2). Thus, all adjacent vertices have distinct weights. Moreover we obtain *χla*(*mC*2*n*+1) ≤ *m* + 2.

Please note that a cycle *C*2*n*+<sup>1</sup> is 1-equally 2-edge colorable. It is possible to generalize the results from the previous section also for *c*(*G*)-equally 2-edge colorable graphs. If we are able to guarantee that for every edge *uv* ∈ *E*(*G*) is

$$\begin{aligned} mwt\_f(u) + (i - m)n\_c^1(u) + (1 - i)n\_c^2(u) \\ \neq mwt\_f(v) + (i - m)n\_c^1(v) + (1 - i)n\_c^2(v) \end{aligned} \tag{3}$$

then we can prove that

$$
\chi\_{la}(m\mathcal{G}) \le \chi\_{la}(\mathcal{G}) + \min\{(m-1)c(\mathcal{G}) : \mathcal{c} \text{ is a 2-edge coloring of } \mathcal{G} \text{ satisfying (3)}\}.
$$

This condition is fulfilled for some graphs containing pendant vertices, thus also for some trees.

**Lemma 2.** *Let G be a graph with l leaves, l* ≥ 0*. Then*

$$
\chi\_{la}(G) \ge l+1.
$$

**Proof.** The proof is similar to the proof in [14]. Let *f* be any *local antimagic labeling* of a graph *G*. Then in the coloring induced by *f* , the color of a leaf *v* is *f*(*uv*), where *uv* ∈ *E*(*G*). Hence all the leaves receive distinct colors. Moreover, for any non-leaf *w* incident with an edge *e* with *f*(*e*) = |*E*(*G*)|, the color assigned to *w* is larger than |*E*(*G*)|. Hence the number of colors in the coloring induced by *f* is at least *l* + 1.

**Theorem 5.** *Let G be a graph without a component isomorphic to K*<sup>2</sup> *such that all vertices in G but leaves have the same even degree. If there exists a* 2*-edge coloring c of G such that for all vertices v but leaves holds n*<sup>1</sup> *<sup>c</sup>* (*v*) = *n*<sup>2</sup> *<sup>c</sup>* (*v*) = deg(*v*)/2*, then*

$$m l + 1 \le \chi\_{la}(m \mathbf{G}) \le \chi\_{la}(\mathbf{G}) + (m - 1)l\_{\prime}$$

*where m is a positive integer and l is the number of leaves in G.*

**Proof.** Let *G* be a graph without a component isomorphic to *K*<sup>2</sup> such that all its vertices but leaves have the same even degree 2*r*. Let *c* be a 2-edge coloring of *G* such that for all vertices *v* in *G* but leaves holds *n*<sup>1</sup> *<sup>c</sup>* (*v*) = *n*<sup>2</sup> *<sup>c</sup>* (*v*) = deg(*v*)/2 = *r*.

Let *f* be any *local antimagic labeling* of a graph *G* that uses *χla*(*G*) colors. Then using Equality (1) we obtain that there exists an edge labeling *g* of *mG*, *m* ≥ 1, such that the weights of non-leaf vertices *vi*, *i* = 1, 2, . . . , *m*, corresponding to a vertex *v* in *G*, are

$$wt\_{\mathcal{S}}(v\_i) = m \cdot wt\_f(v) + (1 - m)r.s$$

This means that the weights of corresponding non-leaf vertices in every copy of *G* are the same. However, this also means that the adjacent non-leaf vertices in *mG* have distinct weights.

Now consider the edges *wiui*, *i* = 1, 2, ... , *m*, where *w* is a leaf. For *i* = 1, 2, ... , *m* trivially holds

$$wt\_{\mathcal{S}}(w\_i) = \mathcal{g}(w\_i \mu\_i) < \sum\_{\iota v \in E(G)} \mathcal{g}(v\_i \mu\_i) = wt\_{\mathcal{S}}(\mu\_i).$$

Which means that all adjacent vertices have distinct weights. Combining the previous arguments we obtain

$$
\chi\_{la}(mG) \le \chi\_{la}(G) + (m-1)l.
$$

The lower bound for *χla*(*mG*) follows from Lemma 2.

**3. Trees**

If the graph in Theorem 5 is a forest we immediately obtain the following result.

**Theorem 6.** *Let T be a forest with no component isomorphic to K*<sup>2</sup> *such that all vertices but leaves have the same even degree. Then*

$$ml + 1 \le \chi\_{la}(mT) \le \chi\_{la}(T) + (m - 1)l\_{\prime\prime}$$

*where m is a positive integer and l is the number of leaves in T.*

**Proof.** Trivially, any graph containing *K*<sup>2</sup> as a component cannot be local antimagic. Let *T* be a forest with no component isomorphic to *K*<sup>2</sup> such that all vertices but leaves have the same even degree 2*r*. Clearly there exists a 2-edge coloring *c* of *T* such that for all vertices *v* but leaves hold *n*<sup>1</sup> *<sup>c</sup>* (*v*) = *n*<sup>2</sup> *<sup>c</sup>* (*v*) = deg(*v*)/2 = *r*. Thus, according to Theorem 5 we are done.

Immediately from the previous theorem we obtain the result for copies of paths and copies of some stars as *χla*(*Pn*) = 3 for *n* ≥ 3 and *χla*(*K*1,*n*) = *n* + 1 for *n* ≥ 2, see [14].

**Corollary 2.** *Let Pn be a path on n vertices, n* ≥ 3*. Then for every positive integer m, m* ≥ 1*, holds*

$$
\chi\_{la}(mP\_n) = 2m + 1.
$$

**Corollary 3.** *Let K*1,2*<sup>n</sup> be a star, n* ≥ 1*. Then for every positive integer m, m* ≥ 1*, holds*

$$
\chi\_{la}(mK\_{1,2n}) = 2nm + 1.
$$

**Theorem 7.** *Let K*1,2*n*+<sup>1</sup> *be a star, n* ≥ 1*. Then for every positive integer m, m* ≥ 1*, holds*

$$\chi\_{la}(m\mathcal{K}\_{1,2n+1}) = \begin{cases} (2n+1)m+1, & \text{if } m \text{ is odd or if } m \text{ is even and } m \ge n+1, \\ (2n+1)m+2, & \text{if } m \text{ is even and } m < n+1. \end{cases}$$

**Proof.** Let us denote the vertices and the edges of *mK*1,2*n*+<sup>1</sup> such that

$$\begin{aligned} V(mK\_{1,2n+1}) &= \{w\_i, v\_i^j : i = 1, 2, \dots, m, j = 1, 2, \dots, 2n+1\}, \\ E(mK\_{1,2n+1}) &= \{w\_i v\_i^j : i = 1, 2, \dots, m, j = 1, 2, \dots, 2n+1\}. \end{aligned}$$

We consider two cases according to the parity of *m*. *Case 1:* when *m* is odd.

We define an edge labeling *g* of *mK*1,2*n*+<sup>1</sup> in the following way

$$g(w\_{l}v\_{i}^{j}) = \begin{cases} i\_{\prime} & \text{if } j = 1 \text{ and } i = 1, 2, \dots, m\_{\prime} \\ \frac{3m+1}{2} + i\_{\prime} & \text{if } j = 2 \text{ and } i = 1, 2, \dots, \frac{m-1}{2}, \\ \frac{m+1}{2} + i\_{\prime} & \text{if } j = 2 \text{ and } i = \frac{m+1}{2}, \frac{m+3}{2}, \dots, m\_{\prime} \\ 3m+1 - 2i\_{\prime} & \text{if } j = 3 \text{ and } i = 1, 2, \dots, \frac{m-1}{2}, \\ 4m+1 - 2i\_{\prime} & \text{if } j = 3 \text{ and } i = \frac{m+1}{2}, \frac{m+3}{2}, \dots, m\_{\prime} \\ (j-1)m+i\_{\prime} & \text{if } j = 4, 5, \dots, n+2 \text{ and } i = 1, 2, \dots, m\_{\prime} \\ jm+1 - i\_{\prime} & \text{if } j = n+3, n+4, \dots, 2n+1 \text{ and } i = 1, 2, \dots, m. \end{cases}$$

Evidently *g* is a bijection and the induced weights of the vertices *wi*, *i* = 1, 2, . . . , *m*, are

$$wt\_{\mathcal{S}}(w\_i) = \sum\_{j=1}^{2n+1} \mathcal{g}(w\_i v\_i^j) = \frac{(2n+1)(m(2n+1)+1)}{2}.$$

As all vertices of degree 2*n* + 1 have the same weights and the weights of the leaves are distinct we obtain *χla*(*mK*1,2*n*+1) ≤ (2*n* + 1)*m* + 1. The lower bound follows from Lemma 2. *Case 2:* when *m* is even.

In this case consider a labeling *f* of *mK*1,2*n*+<sup>1</sup> defined such that

$$f(w\_i v\_i^j) = \begin{cases} j\_\prime & \text{if } j = 1, 2, \dots, 2n + 1 \text{ and } i = 1, \\ 2n + 1 + g(w\_{i-1} v\_{i-1}^j), & \text{if } j = 1, 2, \dots, 2n + 1 \text{ and } i = 2, 3, \dots, m. \end{cases}$$

According to the properties of the labeling *g*, the labeling *f* is a bijective mapping that assigns numbers 1, 2, ... , *m*(2*n* + 1) to the edges of *mK*1,2*n*<sup>+</sup>1. The weights of vertices *wi*, *i* = 2, 3, . . . , *m*, are all the same as

$$wt\_f(w\_i) = \sum\_{j=1}^{2n+1} f(w\_i v\_j^j) = \sum\_{j=1}^{2n+1} \left[ 2n + 1 + g(w\_{i-1} v\_{i-1}^j) \right] = (2n+1)^2 + \frac{(2n+1)(n(2n+1)+1)}{2}.$$

The weight of the vertex *w*<sup>1</sup> is

$$wt\_f(w\_1) = \sum\_{j=1}^{2n+1} f(w\_1 v\_1^j) = \sum\_{j=1}^{2n+1} j = (n+1)(2n+1).$$

If the weight of the vertex *w*<sup>1</sup> under the labeling *f* is the same as the weight of some leaf, we obtain that *χla*(*mK*1,2*n*+1) ≤ (2*n* + 1)*m* + 1. This is satisfied when (*n* + 1)(2*n* + 1) ≤ *m*(2*n* + 1), that is if *n* + 1 ≤ *m*. The equality *χla*(*mK*1,2*n*+1)=(2*n* + 1)*m* + 1 holds because the number of induced colors must be greater then the number of leaves, see Lemma 2. Now consider that the weight of the vertex *w*<sup>1</sup> under the labeling *f* is greater then the weight of all leaves, i.e., *n* + 1 > *m*. Then labeling *f* induces (2*n* + 1)*m* + 2 colors for vertices. To prove that it is not possible to obtain (2*n* + 1)*m* + 1 colors it is sufficient to consider the fact, that the weight of any vertex of degree 2*n* + 1 is at least the sum of numbers 1, 2, ... , 2*n* + 1, thus it is at least (*n* + 1)(2*n* + 1). However, the weights of leaves are at most (2*n* + 1)*m*. Thus if there exists an edge labeling that induces (2*n* + 1)*m* + 1 colors for vertices, under this labeling all vertices *wi*, *i* = 1, 2, ... , *m* must have the same color/weight, say *c*(*w*). However, in this case the sum of all edge labels must be equal to *m* multiple of *c*(*w*), as every edge label contributes exactly once the weight of a vertex of degree 2*n* + 1. Thus *mc*(*w*) = 1 + 2 + ··· + (2*n* + 1)*m* which implies

$$2c(w) = (2n+1)((2n+1)m+1)\dots$$

However, this is a contradiction as for *m* even the right side of the previous equation is odd. This means that in this case *χla*(*mK*1,2*n*+1)=(2*n* + 1)*m* + 2.

Please note that Theorem 6 can be extended also for other trees (forests) such that their non-leaf vertices have even degrees, not necessarily the same. We just need to guarantee that the adjacent non-leaf vertices will have distinct weights. For some trees, for example for spiders, we are able to do it. A *spider graph* is a tree with exactly one vertex of degree greater than 2. By *S*(*n*1, *n*2, ... , *nl*), 1 ≤ *ni* ≤ *ni*+1, *i* = 1, 2, ... , *l* − 1, *l* ≥ 3, we denote a spider obtained by identifying one leaf in paths *Pni*+1, *i* = 1, 2, ... , *l*. In [17] was proved that if *n*<sup>1</sup> = 1 then *χla*(*S*(*n*1, *n*2, ... , *nl*)) = *l* + 1 and if *n*<sup>1</sup> ≥ 2 then *χla*(*S*(*n*1, *n*2, ... , *nl*)) ≤ *l* + 2. Moreover, for *l* ≥ 4 the described edge labeling induces for the root vertex, the vertex of degree *l*, the largest weight over all other vertex weights. Using the presented results we obtain

**Theorem 8.** *Let S*(*n*1, *n*2,..., *nl*) *be a spider graph. If l is even, l* ≥ 4*, and n*<sup>1</sup> = 1

$$\chi\_{la}(m\mathcal{S}(n\_1, n\_2, \dots, n\_l)) = ml + 1.$$

*If l is even, l* ≥ 4*, and n*<sup>1</sup> ≥ 2

$$m l + 1 \le \chi\_{la}(m S(n\_1, n\_2, \dots, n\_l)) \le m l + 2.$$

In [18] was proposed the following conjecture.

**Theorem 9.** *Ref.* [18] *Let T be a tree other than K*<sup>2</sup> *with l leaves. Then*

$$l+1 \le \chi\_{la}(T) \le l+2.$$

In the light of the previous results trees, for copies of trees we conjecture

**Theorem 10.** *Let T be a tree other than K*<sup>2</sup> *with l leaves. Then for every positive integer m, m* ≥ 1*,*

$$ml + 1 \le \chi\_{la}(mT) \le ml + 2.$$

#### **4. Graphs with Chromatic Index 3**

In this section we will deal with 3-regular graphs that admit a proper 3-edge coloring.

**Theorem 11.** *Let G be a* 3*-regular graph with chromatic index χ* (*G*) = 3*. Then for every odd positive integer m, m* ≥ 1*, holds*

$$
\chi\_{la}(m\mathcal{G}) \le \chi\_{la}(\mathcal{G}).
$$

**Proof.** Let *c* be a proper 3-edge coloring of *G*. Let *f* be a local vertex antimagic edge labeling of *G* that uses *χla*(*G*) colors.

We define a new labeling *g* of *mG*, for *m* odd, in the following way.

$$g(e\_i) = \begin{cases} m(f(e) - 1) + i, & \text{if } c(e) = c\_1 \text{ and } i = 1, 2, \dots, m, \\ m(f(e) - 1) + i + \frac{m + 1}{2}, & \text{if } c(e) = c\_2 \text{ and } i = 1, 2, \dots, \frac{m - 1}{2}, \\ m(f(e) - 1) + i - \frac{m - 1}{2}, & \text{if } c(e) = c\_2 \text{ and } i = \frac{m + 1}{2}, \frac{m + 3}{2}, \dots, m, \\ mf(e) + 1 - 2i, & \text{if } c(e) = c\_3 \text{ and } i = 1, 2, \dots, \frac{m - 1}{2}, \\ mf(e) + m + 1 - 2i, & \text{if } c(e) = c\_3 \text{ and } i = \frac{m + 1}{2}, \frac{m + 3}{2}, \dots, m. \end{cases}$$

It is easy to see that if an edge in *G* is labeled with the number *t*, 1 ≤ *t* ≤ |*E*(*G*)|, then the corresponding edges in *mG* are labeled with numbers from the set {*m*(*t* − 1) + 1, *m*(*t* − 1) + 2, ... , *mt*}. Thus, *g* is a bijection that assigns numbers 1, 2, ... , *m*|*E*(*G*)| to the edges of *mG*. Now we will calculate a vertex weight of the vertex *vi* in *mG*. Let *x*, *y* and *z* be the vertices adjacent to *v* in *G*. Without loss of generality we can assume that *c*(*xv*) = *c*1, *c*(*yv*) = *c*<sup>2</sup> and *c*(*zv*) = *c*3. Then for *i* = 1, 2, . . . ,(*m* − 1)/2 we obtain

$$\begin{split} wt\_{\mathcal{g}}(v\_{i}) &= \mathbf{g}(\mathbf{x}\_{i}v\_{i}) + \mathbf{g}(y\_{i}v\_{i}) + \mathbf{g}(z\_{i}v\_{i}) \\ &= [m(f(\mathbf{x}v) - 1) + i] + \left[ m(f(yv) - 1) + i + \frac{m+1}{2} \right] + [mf(zv) + 1 - 2i] \\ &= m(f(\mathbf{x}v) + f(yv) + f(zv)) + \frac{3-3m}{2} = mwt\_{f}(v) + \frac{3-3m}{2}. \end{split}$$

If *i* = (*m* + 1)/2,(*m* + 3)/2, . . . , *m* then

$$\begin{split} wt\_{\mathcal{S}}(v\_{i}) &= g(\mathbf{x}\_{i}v\_{i}) + g(y\_{i}v\_{i}) + g(z\_{i}v\_{i}) \\ &= [m(f(\mathbf{x}v) - 1) + i] + \left[ m(f(yv) - 1) + i - \frac{m-1}{2} \right] + [mf(zv) + m + 1 - 2i] \\ &= m(f(\mathbf{x}v) + f(yv) + f(zv)) + \frac{3-3m}{2} = mwt\_{f}(v) + \frac{3-3m}{2}. \end{split} \tag{4}$$

Thus, in all copies the corresponding vertices have the same weights. Moreover, as the set of weights of vertices in *G* under the labeling *f* consists of *χla*(*G*) distinct numbers we immediately obtain that also the set of weights of vertices in *mG* under the labeling *g* consists of *χla*(*G*) distinct numbers. Thus, *χla*(*mG*) ≤ *χla*(*G*).

Analogously, as it was possible to extend the results in Section 2 for graphs with leaves, we can also extend Theorem 11 for some graphs with pendant vertices.

**Theorem 12.** *Let G be a graph such that all vertices but leaves have degree* 3*. If there exists a* 3*-edge coloring c of G such that for all vertices v but leaves hold n*<sup>1</sup> *<sup>c</sup>* (*v*) = *n*<sup>2</sup> *<sup>c</sup>* (*v*) = *n*<sup>3</sup> *<sup>c</sup>* (*v*) = 1*, then for every odd positive integer m, m* ≥ 1*,*

$$m l + 1 \le \chi\_{la}(m \mathbf{G}) \le \chi\_{la}(\mathbf{G}) + (m - 1)l\_{\prime}$$

*where l is the number of leaves in G.*

**Proof.** Let *G* be a graph such that all its vertices but leaves have degree 3. Let *c* be a 3-edge coloring of *G* such that for all vertices *v* in *G* but leaves hold *n*<sup>1</sup> *<sup>c</sup>* (*v*) = *n*<sup>2</sup> *<sup>c</sup>* (*v*) = *n*<sup>3</sup> *<sup>c</sup>* (*v*) = 1. Let *f* be any *local antimagic labeling* of a graph *G* that uses *χla*(*G*) colors. Then using Equality (4) we obtain that there exists an edge labeling *g* of *mG*, *m* odd *m* ≥ 1, such that the weights of non-leaf vertices *vi*, *i* = 1, 2, . . . , *m*, corresponding to a vertex *v* in *G*, are

$$wt\_{\mathcal{S}}(v\_i) = mwt\_f(v) + \frac{3-3m}{2}.$$

This means that the weights of corresponding non-leaf vertices in every copy of *G* are the same. However, this also means that the adjacent non-leaf vertices in *mG* have distinct weights.

Now consider the edges *wiui*, *i* = 1, 2, . . . , *m*, where *w* is a leaf. Trivially holds

$$wt\_{\mathcal{S}}(w\_i) = \mathcal{g}(w\_i u\_i) < \sum\_{\iota v \in E(G)} \mathcal{g}(v\_i u\_i) = wt\_{\mathcal{S}}(u\_i).$$

Which means that all adjacent vertices have distinct weights. Combining the previous arguments we obtain

$$
\chi\_{la}(mG) \le \chi\_{la}(G) + (m-1)l.
$$

The lower bound for *χla*(*mG*) follows from Lemma 2.

Immediately for forests we obtain the following result.

**Corollary 4.** *Let T be a forest such that all its vertices but leaves have degree* 3*. Then for every odd positive integer m, m* ≥ 1 *holds*

$$ml + 1 \le \chi\_{la}(mT) \le \chi\_{la}(T) + (m - 1)l\_{\prime\prime}$$

*where l is the number of leaves in T.*

**Proof.** Let *T* be a forest such that all its vertices but leaves have degree 3. Trivially there exists a 3-edge coloring *c* of *T* such that for all vertices *v* but leaves hold *n*<sup>1</sup> *<sup>c</sup>* (*v*) = *n*<sup>2</sup> *<sup>c</sup>* (*v*) = *n*3 *<sup>c</sup>* (*v*) = 1. Using Theorem 12 we obtain the desired result.

The next theorem shows how it is possible to extend the previous result for regular graphs that are decomposable into spanning subgraphs that are all isomorphic either to even regular graphs or 3-regular graphs.

**Theorem 13.** *Let G be a graph that can be decomposed into factors G*1, *G*2, ... , *Gk, k* ≥ 1*, and let every factor Gi, i* = 1, 2, . . . , *k, be isomorphic to a graph of the following types:*

*type I: a* 4*-regular graph, type II: a* 2*-regular graph consisting of even cycles, type III: a* 3*-regular graph with chromatic index* 3*.*

*If every factor Gi, i* = 1, 2, ... , *k, is of type I or of type II then for every positive integer m, m* ≥ 1*, holds*

$$
\chi\_{la}(m\mathbf{G}) \le \chi\_{la}(\mathbf{G}).
$$

*If at least one factor Gi, i* = 1, 2, ... , *k, is of type III then for every odd positive integer m, m* ≥ 1*, holds*

$$
\chi\_{la}(m\mathbf{G}) \le \chi\_{la}(\mathbf{G}).
$$

Please note that the exact value of *χla*(*Kn*) is *n*, since *χla*(*Kn*) ≥ *χ*(*Kn*) = *n*. Immediately from the previous theorem we obtain the following result for complete graphs *Kn*.

**Corollary 5.** *Let Kn be a complete graph on n vertices, n* ≥ 4*. If n* ≡ 1 (mod 4) *then for every positive integer m, m* ≥ 1*, and if n* ≡ 0 (mod 4) *then for every odd positive integer m, m* ≥ 1*, we have χla*(*mKn*) = *n.*

#### **5. Conclusions**

One interesting problem is to find a local antimagic chromatic number for disjoint union of arbitrary graphs. According to results proved by Haslegrave [16] we obtain that this parameter is finite for disjoint union of arbitrary graphs if and only if non of these graphs contains an isolated edge as a subgraph. Moreover, Haslegrave [16] proved the following result.

**Theorem 14.** *Ref.* [16] *For every graph G with m edges, none of which is isolated, and for any positive integer k, the edges of G may be labeled with a permutation of* {*k*, *k* + 1, ... , *m* + *k* − 1} *in such a way that the vertex sums distinguish all pairs of adjacent vertices.*

Immediately from this result we obtain an upper bound for a local antimagic chromatic number for disjoint union of arbitrary graphs.

**Theorem 15.** *Let Gi, i* = 1, 2, . . . , *n, be a graph with no isolated edge. Then*

$$\chi\_{la}\left(\bigcup\_{i=1}^{n}\mathcal{G}\_{l}\right) \le \min\left\{\chi\_{la}(\mathcal{G}\_{l}) + \sum\_{i=1}^{n} |V(\mathcal{G}\_{l})| - |V(\mathcal{G}\_{l})| \, : \, t = 1, 2, \dots, n\right\}.$$

For some graphs we can obtain a better upper bound.

**Theorem 16.** *Let Gi, i* = 1, 2*, be a graph with no isolated edge. Let G*<sup>2</sup> *be a graph such that all vertices but leaves have the same degree. Then*

$$
\chi\_{la}(\mathcal{G}\_1 \cup \mathcal{G}\_2) \le \chi\_{la}(\mathcal{G}\_1) + \chi\_{la}(\mathcal{G}\_2) + l\_{2\prime}
$$

*where l*<sup>2</sup> *is the number of leaves in G*2*.*

**Proof.** Let *G*<sup>2</sup> be a graph such that all vertices but leaves have the same degree *r*, *r* ≥ 2 and let *l*<sup>2</sup> be the number of leaves in *G*2. Let *fi*, *i* = 1, 2, be a local vertex antimagic edge labeling of *Gi* that uses *χla*(*Gi*) colors. We define an edge labeling *g* of *G*<sup>1</sup> ∪ *G*<sup>2</sup> such that

$$g(e) = \begin{cases} f\_1(e)\_\prime & \text{if } e \in E(G\_1)\_\prime \\ f\_2(e) + |E(G\_1)|\_\prime & \text{if } e \in E(G\_2)\_\prime \end{cases}$$

As *f*<sup>1</sup> and *f*<sup>2</sup> are bijections evidently also *g* is a bijection. For the vertex weights under the labeling *g* we obtain the following. If *v* ∈ *V*(*G*1) then

$$wt\_{\mathcal{S}}(\upsilon) = \sum\_{\iota\upsilon\upsilon \in E(G\_1)} \mathcal{g}(\iota\upsilon) = \sum\_{\iota\upsilon\upsilon \in E(G\_1)} f\_1(\iota\upsilon\upsilon) = wt\_{f\_1}(\upsilon).$$

Thus, the weights of adjacent vertices in *G*<sup>1</sup> are distinct and they induce *χla*(*G*1) colors. If *v* ∈ *V*(*G*2) and deg*G*<sup>2</sup> (*v*) = *r* then

$$\begin{aligned} wt\_{\mathcal{S}}(v) &= \sum\_{\iota v \in E(G\_2)} g(\iota v) = \sum\_{\iota v \in E(G\_2)} (f\_2(\iota v) + |E(G\_1)|) \\ &= \sum\_{\iota v \in E(G\_2)} f\_2(\iota v) + r|E(G\_1)| = wt\_{f\_2}(v) + r|E(G\_1)|. \end{aligned}$$

If *v* ∈ *V*(*G*2) and deg*G*<sup>2</sup> (*v*) = 1 then

$$\begin{aligned} wt\_{\mathcal{S}}(\upsilon) &= \sum\_{uv \in E(\mathcal{G}\_2)} g(uv) = \sum\_{uv \in E(\mathcal{G}\_2)} (f\_2(uv) + |E(\mathcal{G}\_1)|) = f\_2(uv) + |E(\mathcal{G}\_1)|. \\ &= wt\_{f\_2}(\upsilon) + |E(\mathcal{G}\_1)|. \end{aligned}$$

This means that also the weights of adjacent vertices in *G*<sup>2</sup> are distinct. Moreover, we obtain that the labeling *g* induces at most *χla*(*G*2) + *l*<sup>2</sup> colors as the number of colors assigned to the vertices of degree at least 2 is the same and all the leaves could be assigned with the colors different from the colors of non leaves.

Combining the previous we obtain that the labeling *g* induces at most *χla*(*G*1) + *χla*(*G*2) + *l*<sup>2</sup> colors.

**Theorem 17.** *Let G be a graph with no isolated edge and with l leaves. Then for every positive integer m, m* ≥ 1 *holds*

$$l + 2m + 1 \le \chi\_{la}(G \cup mP\_{\mathfrak{J}}) \le \chi\_{la}(G) + 2m + 1.$$

**Proof.** Let *G* be a graph with no isolated edge and with *l* leaves. The lower bound follows from Lemma 2. The upper bound is based on the fact that there exists a *local antimagic labeling* of *mP*<sup>3</sup> that induces 2*m* + 1 colors such that the color of every vertex of degree 2 in *mP*<sup>3</sup> will have the same color 2*m* + 1. Thus, the labeling *g* of *mP*<sup>3</sup> ∪ *G* described in the proof of Theorem 16 induces also 2*m* + 1 colors for vertices in *mP*3. These colors are |*E*(*G*)| + 1, |*E*(*G*)| + 2, ... , |*E*(*G*)| + 2*m* and 2|*E*(*G*)| + 2*m* + 1. In general, these colors are distinct from colors of vertices in *G*<sup>1</sup> induced by the labeling *g*. This concludes the proof.

**Author Contributions:** Conceptualization, M.B., A.S.-F. and T.-M.W.; methodology, M.B., A.S.-F. and T.-M.W.; validation, M.B., A.S.-F. and T.-M.W.; investigation, M.B., A.S.-F. and T.-M.W.; resources, M.B., A.S.-F. and T.-M.W.; writing—original draft preparation, A.S.-F.; writing—review and editing, M.B., A.S.-F. and T.-M.W.; supervision, A.S.-F.; project administration, M.B., A.S.-F. and T.-M.W.; funding acquisition, M.B., A.S.-F. and T.-M.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Slovak Research and Development Agency under the contract No. APVV-19-0153 and by VEGA 1/0233/18. Also for the author Tao-Ming Wang the research is supported by MOST 108-2115-M-029-002 from the ministry of science and technology of Taiwan.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


#### **José Ángel Juárez Morales 1, Gerardo Reyna Hernández 1,\*, Jesús Romero Valencia <sup>2</sup> and Omar Rosario Cayetano <sup>1</sup>**


**Abstract:** Often for understanding a structure, other closely related structures with the former are associated. An example of this is the study of hyperspaces. In this paper, we give necessary and sufficient conditions for the existence of finitely-dimensional maximal free cells in the hyperspace *C*(*G*) of a dendrite *G*; then, we give necessary and sufficient conditions so that the aforementioned result can be applied when *G* is a dendroid. Furthermore, we prove that the arc is the unique arcwise connected, compact, and metric space *X* for which the anchored hyperspace *Cp*(*X*) is an arc for some *p* ∈ *X*.

**Keywords:** hyperspace; graph; dendroid; dendrite

#### **1. Introduction**

In the study of a mathematical structure, sometimes other structures that allow for visualizing problems in different ways are built.

One of the theories developed using this type of study is the Theory of Hyperspaces; this theory began with the investigations of F. Hausdorff and L. Vietoris. Given a topological space *X*, the 2*<sup>X</sup>* hyperspace of all nonempty and closed subsets of *X* was introduced by L. Vietoris in 1922, and he proved basic facts about 2*X*—for example, compactness of *X* implies compactness of 2*<sup>X</sup>* and vice versa; 2*<sup>X</sup>* is connected if and only if *X* is connected. When *X* is a metric space, 2*<sup>X</sup>* can be endowed with the Hausdorff metric (defined by F. Hausdorff in 1914).

The hyperspace of all nonempty, closed and connected subsets of *X* is denoted by *C*(*X*) and considered as a subspace of 2*X*. In turn, the hyperspace of all nonempty, closed, and connected sets of *X* containing a point *p*, which is denoted by *Cp*(*X*), is a subspace of *C*(*X*).

The hyperspaces *C*(*X*) and *Cp*(*X*) are subjects of study for many researchers. Among several topics about hyperspace, one of the most interesting is to recognize a hyperspace as homeomorphic to some known space: Ref. [1] presents a special class of spaces *X* for which *C*(*X*) is homeomorphic to the infinite cylinder *X* × R≥0. Another interesting topic is to analyze topological properties: for compact, connected, and metric *X*, the hyperspaces *Cp*(*X*) are locally connected for all *p* ∈ *X* [2].

Graphs have been widely and deeply studied (see [3–7]) and have proved to be an excellent tool for representing and modeling different structures in several areas of discrete mathematics and computation (see [8,9]). As far as hyperspace is concerned, there exist some works relating both subjects. For example, Duda [10] proved that a space *X* is a finite graph if and only if *C*(*X*) is a polyhedral. In a dendroid *X* smooth in a point *p*, *Cp*(*X*) is homeomorphic to the Hilbert cube if and only if *p* is not in the interior of a finite tree in *X*, a result due to Carl Eberhart [11]. Recently, Reyna et al. proved that, in a local space *X*, *Cp*(*X*) is a polyhedral for all *p* if and only if *X* is a finite graph [12].

In this paper, we are concerned with fully determining the existence of maximal finite dimensional free cells in the hyperspace *C*(*X*), first of a dendrite and then a dendroid *X*,

**Citation:** Morales, J.Á.J.; Hernández, G.R.; Valencia, J.R.; Cayetano, O.R. Free Cells in Hyperspaces of Graphs. *Mathematics* **2021**, *9*, 1627. https:// doi.org/10.3390/math9141627

Academic Editors: Janez Žerovnik and Darja Rupnik Poklukar

Received: 22 April 2021 Accepted: 8 July 2021 Published: 10 July 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

as well as examining necessary and sufficient conditions for the hyperspace *Cp*(*X*) if an arc provided *X* is an arc-wise connected space.

#### **2. Definitions**

Throughout this paper, the term *space* is meant to be a connected, compact, and metric space, and a *subspace* is understood to be a subset of a space which is a space itself. Given a space *X*, the symbol 2*<sup>X</sup>* denotes the *hyperspace* of non-empty closed subsets of *X*, and *C*(*X*) is the hyperspace of non empty subspaces of *X* both endowed with the *Hausdorff's metric*, two models of hyperspaces are shown in Figure 1. Notice that *X* is naturally embedded in <sup>2</sup>*<sup>X</sup>* via the map *<sup>x</sup>* → {*x*} (compare with ([13] [0.48]).

**Figure 1.** The hyperspaces *C*(*M*) for the path *P*<sup>2</sup> and the star *S*3.

Given a point *p* ∈ *X*, the *anchored hyperspace* of *X* at *p*, denoted by *Cp*(*X*), is the subspace of *C*(*X*) consisting of those elements containing *p*. Note that *Cp*(*X*) is a subspace of *C*(*X*), which in turn is a subspace of 2*X*.

A space *X* is *unicoherent* if, for any *A*, *B* ⊂ *X* subspaces such that *X* = *A* ∪ *B*, we have that *A* ∩ *B* is connected. The space *X* is called *hereditary unicoherent* if each subspace is unicoherent.

A *graph G*, consisting of a set *V*(*G*), called the *vertices* of *G* and a set *E*(*G*) of unordered pairs of elements of *V*(*G*), called the *edges* of *G*. Letting *G* be a graph, if two vertices *x* and *y* of *G* form an edge, we say that they are *adjacent*, and this is denoted by *x* ∼ *y*. This fact is also expressed by saying that *x* and *y* are *neighbors*. A vertex of *G* is called a *ramification vertex* if it has three or more neighbors and a *terminal vertex* if it has exactly one neighbor. *G* is called *simple* if it contains no loops (a vertex adjacent to itself) and possesses at most one edge between any two vertices. A *path* between two vertices *u* and *v* of *G* is a finite sequence of consecutive adjacent vertices such that the first one is *u* and the last one is *v*. *G* is *connected* if there is a path between any two vertices. A *cycle* in *G* is a finite sequence of at least three consecutively adjacent vertices such that the first one and the last one are adjacent. In this paper, we consider simple and connected graphs without cycles whose vertices are ramification or terminal vertices, that is, there are no vertices with exactly two neighbors.

In order to consider a graph *G* as a metric space, if we use the notation [*u*, *v*] for the edge joining the vertices *u* and *v*, we must identify any edge [*u*, *v*] ∈ *E*(*G*) with the closed interval [0, *l*] (if *l* := *L*([*u*, *v*]); therefore, any point in the interior of any edge is a point of *G* and, if we consider the edge [*u*, *v*] as a graph with just one edge, then it is identified with the closed interval [0, *l*]. A connected graph *G* is naturally equipped with a distance defined on its points, induced by taking shortest paths in *G*. Then, we see *G* as a metric graph (see [10,14]); according to this, a *dendroid* is a simple and connected graph without cycles which is a hereditary unicoherent space; the *comb* and the *harmonic fan* are examples of dendroids (see Figure 2). By *dendrite*, we mean a locally connected dendroid. Any tree, the *Fω*, and the *Gehmann* dendrite are examples of these types of graphs (see Figure 3). Throughout this paper, *G* denotes a dendroid or a dendrite.

**Figure 2.** The comb and harmonic fan dendroids.

**Figure 3.** The *Fω* and Gehmann dendrites.

A point *p* ∈ *G* is called *essential of type I* if it is a vertex with infinitely many neighbors or *essential of type II* if there exists an infinite sequence of ramification vertices (*pn*) such that *pn* → *p*. We use the word *essential* to mean essential of type I or II. A point which is not a vertex, nor an essential point, is called an *ordinary point*; we denote *T*(*G*), *O*(*G*), *R*(*G*), and *ES*(*G*) the sets of terminal vertices, ordinary points, ramification vertices, and essential points, respectively.

The *order* of a point *x* in a dendroid *G* is defined as follows:

$$o\_G(\mathfrak{x}) = \begin{cases} 1, & \text{if } \mathfrak{x} \text{ is a terminal vertex;}\\ 2, & \text{if } \mathfrak{x} \text{ is an ordinary point;}\\ n, & \text{if } \mathfrak{x} \text{ has exactly } n \text{ neighbors;}\\ \infty, & \text{if } \mathfrak{x} \text{ is an essential point.} \end{cases}$$

An *m-dimensional cell* (or *m-cell* for short) in a space *X* is a subspace *M* homeomorphic to [0, 1] *<sup>m</sup>*, the part of *M* homeomorphic to (0, 1)*<sup>m</sup>* is called the *interior manifold* of *M*, and it is denoted by *M*◦, while *M* − *M*◦ is denoted with *∂M*, and it is called the *boundary manifold* of *M*. If it occurs that the interior manifold *M*◦ is actually an open set of *X*, then *M* is called a *free cell* of *X*; Figure 4 shows a space with some free cells. In Theorem 2, we establish sufficient and necessary conditions for the existence of a maximal free *m*-cell in the hyperspace *C*(*G*) for a dendrite *G*. Furthermore, in Theorem 3, we establish sufficient and necessary conditions so that Theorem 2 can be applied when *G* is an arbitrary dendroid.

**Figure 4.** Free cells.

#### **3. Preliminaries**

Given *m* ≤ *n*, let *A* and *B* be *m*, *n*-cells, respectively, with *A* ⊆ *B*. If *m* = *n*, *A*◦ is an open subset of *B*◦. On the other hand, if *m* < *n*, then *A*◦ is not an open subset of *B*◦ because non-empty neighborhoods of *B*◦ contain *m*-dimensional open balls, and none of these can be contained in *A*◦. Therefore, the next lemma follows at once.

**Lemma 1.** *(a) If a cell is contained in a higher dimensional cell, then the first one is not a free cell. (b) Each m-cell contained in a free m-cell is a free m-cell.*

In order to show that the cells that we are going to locate in hyperspace *C*(*G*) of the dendrite *G* are maximal, we need Corollary 1 and Lemmas 2–5; in all of these, except Lemma 3, it is assumed that *A* ⊆ *B* are *m*-cells.

Recall that Int *A* and Bd *A* designate, respectively, the *topological interior* and *topological boundary* of the set *A*.

**Lemma 2.** *If ∂A* = *∂B, then A* = *B.*

**Proof.** Since *A*◦ ⊆ *B*◦, it remains to show that *B*◦ ⊆ *A*◦. Let *x* ∈ *B*◦ and suppose *x* ∈/ *A*◦. Now, if we take *y* ∈ *A*◦, then *x*, *y* ∈ *B*◦. Let *α* be an arc from *x* to *y* contained in *B*◦. Then, the arc *α* contains an end point in *B* − *A* and the other end point in *A*◦. It necessarily occurs that *α* ∩ *∂A* = ∅, and this is absurd.

Recall that the *Borsuk–Ulam* Theorem establishes that, for any continuous map *f* : *<sup>S</sup><sup>n</sup>* <sup>→</sup> <sup>R</sup>*n*, there must exist some point *<sup>x</sup>* <sup>∈</sup> *<sup>S</sup><sup>n</sup>* such that *<sup>f</sup>*(*x*) = *<sup>f</sup>*(−*x*). This theorem, in particular, implies that no such maps can be *one to one*, and this is the key piece in the proof of next lemma.

**Lemma 3.** *The unique homemorphic copy of S<sup>n</sup> contained in S<sup>n</sup> is S<sup>n</sup> itself.*

**Proof.** Let *A* be a proper homeomorphic subspace of *Sn*; notice that we can suppose that the North Pole is not contained in *A* (otherwise, apply a suitable rotation to *Sn*). If *<sup>ψ</sup>* : *<sup>S</sup><sup>n</sup>* <sup>→</sup> *<sup>A</sup>* is a homeomorphism and *<sup>σ</sup>* is the usual stereographic projection of *<sup>S</sup><sup>n</sup>* to <sup>R</sup>*<sup>n</sup>* restricted to *<sup>A</sup>*, then *<sup>σ</sup>* ◦ *<sup>ψ</sup>* : *<sup>S</sup><sup>n</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup>* is a continuous *one to one* map, a contradiction.

**Corollary 1.** *If ∂A* ⊆ *∂B, then A* = *B.*

**Lemma 4.** *If B*◦ ⊆ *A, then A* = *B.*

**Proof.** Let *x* ∈ *∂B* and let *U* be a neighborhood of *x* in *B*. Since *U* ∩ *B*◦ = ∅, *U* ∩ *A* = ∅, and hence *x* ∈ *A* = *A*.

**Lemma 5.** *Let x* ∈ *∂A, if x* ∈/ *∂B, then x* ∈ *Bd A.*

**Proof.** Suppose *x* ∈/ Bd *A*; then, *x* ∈ Int *A* and hence Int *A* ∩ *B*◦ is an open set in *B* containing *x*. Thus, there must exist a neighborhood *V* of *x* homeomorphic to (0, 1)*<sup>m</sup>* contained in Int *A* ∩ *B*◦, and the latter shows that *x* cannot belong to any face of *A*. In other words, *x* ∈/ *∂A*.

If *J* = [*p*1, *p*2] is an arc, it is a well known fact that *C*(*J*) is a 2-cell whose interior manifold are all subsets in the form *A* = [*a*, *b*], where *a* = *b* and none of these points equal to *p*<sup>1</sup> or *p*<sup>2</sup> (see [10]).

#### **4. Free Cells in Hyperspaces of Dendrites**

#### *4.1. The Case n* = 2

We are close to announcing Theorem 2 where necessary and sufficient conditions are given for the existence of a maximal finite dimensional free *n*-cells in the hyperspace *C*(*G*) of a dendrite *G*. The free cell built in its proof has the property that all of their elements contain a certain subspace *A*. In this particular case, the maximal free cells are the hyperspaces *C*(*J*) with *J* an edge, and none of these cells have such a property. Therefore, the case *n* = 2 needs to be treated separately. However, first, it is necessary to state the following known property about locally connected topological spaces.

**Lemma 6.** *In any locally connected topological space, the components of open sets are open sets.*

**Theorem 1.** *The hyperspace C*(*G*) *of a dendrite G contains a maximal free* 2*-cell* B *if and only if* B = *C*(*J*) *for some edge J of G.*

**Proof.** Let *J* be an edge of *G* and *p*1, *p*<sup>2</sup> their extremes, consider an element *A* = [*a*, *b*] ∈ (*C*(*J*))◦ and let *ε* > 0 such that *Nε*(*A*) ⊆ *J*◦ (where *Nε*(*A*) is the union of all open balls *B*(*x*,*ε*) as *x* ranges over all points of *A*). Hence, if *B*(*A*,*ε*) is the open ball (in the Hausdorff metric of *C*(*G*)) centered at *A*, then *B*(*A*,*ε*) ⊆ (*C*(*J*))◦ and *C*(*J*) ⊂ *C*(*G*) is a free 2-cell.

Now, we see that the free cell *C*(*J*) is maximal. Let A be a free 2-cell in *C*(*G*) containing *C*(*J*). If *A* ∈ *∂C*(*J*), we have (i) *A* = [*pj*, *x*], (*j* = 1 *or j* = 2) or else (ii) *A* = {*x*} with *x* ∈ [*p*1, *p*2]. We claim that *A* ∈ *∂*A. If *A* is as (i), we have three sub-cases:

(1) *The point pj is a ramification vertex.* Suppose with no loss of generality that *j* = 1. If *A* ∈/ *∂*A, then *A* ∈ A◦ and hence there must exist *ε* > 0 with *B*(*A*,*ε*) ⊆ A◦. Take *x*1, *x*<sup>2</sup> ∈ *Nε*(*A*) − *A* such that [*x*1, *p*1] ∩ [*x*2, *p*1] = {*p*1}, [*xi*, *p*1] ⊆ *B*(*p*1,*ε*) (*i* ∈ {1, 2}) and a point *u* ∈ (*x*, *p*2) such that [*u*, *x*] ⊆ *B*(*x*,*ε*). The set

$$\mathcal{B} = \{ A \cup [w\_1, p\_1] \cup [w\_2, p\_1] \cup [\mathfrak{x}, w\_3] \mid w\_1 \in [\mathfrak{x}\_1, p\_1], w\_2 \in [\mathfrak{x}\_2, p\_1], w\_3 \in [\mathfrak{u}, \mathfrak{x}] \}$$

is a 3-cell (see [13] [Theorem 1.100]) contained in *B*(*A*,*ε*) ⊆ A◦, and this is absurd. (2) *The point pj is essential.*


The above shows that *A* ∈ *∂*A as desired. For the case ii), if we assume that *A* = {*x*} and *A* ∈/ *∂*A, take *H* ∈ A◦ − *C*(*J*) (see Lemma 4) and notice that *H* does not contain ramification points or essential points; otherwise, in a neighborhood of *H* contained in A◦, 3-cells or even *Hilbert cubes* can be located (in the proof of Theorem 2, it is shown in detail how is this possible). Hence, *H* is an arc and let *q*<sup>1</sup> and *q*<sup>2</sup> denote their end points; according to this, it must be *p*<sup>1</sup> ∈ [*x*, *q*1] or else *p*<sup>2</sup> ∈ [*x*, *q*1]. Suppose *p*<sup>1</sup> ∈ [*x*, *q*1], and, using the fact that *H* = {*x*} and A◦ is arcwise connected, take *α* ⊆ A◦ an arc from {*x*} to *H*. We claim that there exists *L* ∈ *α* such that *p*<sup>1</sup> ∈ *L*. Otherwise, we have *α* ⊆ *C*(*G*) − *Cp*<sup>1</sup> (*G*). Let *U* be the component of *G* − {*p*1} containing *x*, let *V* be the union of the remaining components and notice that *H* ⊆ *V*. By Lemma 6, *U* and *V* are open sets, hence U = {*B* ∈ *C*(*G*) | *B* ⊆ *U*} and V = {*B* ∈ *C*(*G*) | *B* ⊆ *V*} are non-empty, disjoint open sets in *C*(*G*) − *Cp*<sup>1</sup> (*G*) (compare with [15] [Theorem 4.5]) and therefore the sets *α* ∩ U and *α* ∩ V form a separation of *α*, which is impossible, being *α* connected. This proves the existence of the desired *L*.

The point *p*<sup>1</sup> is a ramification vertex or an essential point; since *L* ∈ A◦, as in the sub-cases (1) and (2) for some suitable *ε* > 0, it is possible to find a 3-cell contained in *B*(*L*,*ε* ) ⊆ A◦, and this is a contradiction once again. Therefore, in this case, it must be *A* ∈ *∂*A and the result now follows from Corollary 1.

For the converse, let B ⊆ *C*(*G*) be a maximal free 2-cell and let *A* ∈ B◦. Notice that *A* does not contain ramification vertices or essential points. The above remarks result in *A* needing to be an arc; if *J* = [*p*1, *p*2] (where *p*1, *p*<sup>2</sup> are vertices of *G*) is the edge containing *A*, we claim that B ⊆ *C*(*J*). Otherwise, let *B* ∈B− *C*(*J*). Hence, for each *x* ∈ *B* and for each *y* ∈ *A*, it occurs that a) *p*<sup>1</sup> ∈ [*x*, *y*] or b) *p*<sup>2</sup> ∈ [*x*, *y*]. Suppose without loss of generality that a) occurs and let *α* be an arc in B from *B* to *A* such that *α* − {*B*}⊆B◦. Since *B* ∈ *C*(*J*) and *A* ∈ *C*(*J*), there must exist *C* ∈ *α* ∩ *∂C*(*J*). Hence, *C* has the form [*p*1, *a*]. If *ε* > 0 is such that *B*(*C*,*ε*) ⊆ B◦, then *B*(*C*,*ε*) contains a 3-cell (if *p*<sup>1</sup> is a ramification point) or even a Hilbert cube (if *p*<sup>1</sup> is an essential point). This is a contradiction in any case. This shows that *B* ∈ *C*(*J*) and therefore B = *C*(*J*).

*4.2. The Case n* > 2

We need to introduce some terminology about the hyperspace *C*(*K*) for a tree *K* (for more details, see [10,12]).

An *internal tree T* of a tree *K* is a subgraph which is a tree not containing terminal vertices of *K*. Let *IT*(*K*) denote the set of internal trees of *K*. For *T* ∈ *IT*(*K*), let *I*1, ... , *In* be those edges of *K* such that *Ii* ∩ *T* = ∅ and *Ii* is not contained in *T*. We define

$$D(1,T) = T \cup \left(\bigcup\_{i=1}^n I\_i\right).$$

and we say that this is the *canonical representation* of *D*(1, *T*). Given an internal tree *T* ⊂ *K*, let M(*T*) be the family of all subspaces of *K* in the form

$$((c\_i)\_{i=1}^n)\_T = T \cup \left(\bigcup\_{i=1}^n [\mathbf{0}\_{I\_{i'}} c\_i]\right)\_T$$

where 0*Ii* is the vertex of *Ii* contained in *T*, and [0*Ii* , *ci*] is the subarc of *Ii* joining 0*Ii* with *ci*.

**Lemma 7.** *Let K be a tree, then*


$$\mathcal{C}(K) = \left[ \bigcup\_{T \in IT(K)} \mathfrak{M}(T) \right] \cup \left[ \bigcup\_{I \in E(K)} \mathcal{C}(I) \right].$$

**Theorem 2.** *The hyperspace C*(*G*) *of a dendrite G contains a maximal free n-cell (n* > 2*) if and only if there exists a tree K* ⊆ *G satisfying the following conditions:*


**Proof.** For each *pi* ∈ *T*(*K*), let *ri* ∈ *R*(*K*) such that [*ri*, *pi*] ∩ *R*(*K*) = {*ri*}.

Put *A* = *K* − *<sup>n</sup> i*=1 (*ri*, *pi*] and for each **x** = (*xi*)*<sup>n</sup> i*=1 ∈ *n* ∏ *i*=1 [*ri*, *pi*] let *C***<sup>x</sup>** denote the set *A* ∪ *<sup>n</sup> i*=1 [*ri*, *xi*] . We claim that the family M(*A*) = {*C***<sup>x</sup>** | **x** ∈ *n* ∏ *i*=1 [*ri*, *pi*]} is a maximal free *n*-cell in *C*(*G*).

That M(*A*) is actually a *n*-cell is due to [13], [Theorem 1.100]; therefore, we only need to verify the maximal and free properties.

Let *C***<sup>x</sup>** ∈ (M(*A*))◦ and define *L* = (*G* − *K*) ∪ {*p*1, *p*2, ... , *pn*}. Put *α* = *d*(*C***x**, *L*) = *inf* {*d*(*c*, *l*) | *c* ∈ *C***x**, *l* ∈ *L*}, *α<sup>i</sup>* = *d*(*xi*, *A*), *βij* = *d*(*xi*, [*rj*, *pj*]), where *i* = *j* and *i*, *j* ∈ {1, 2, ... , *n*}. Since all these quantities are positive, take *ε* > 0 less than all those and *Y* ∈ *B*(*C***x**,*ε*). For each *i* ∈ {1, 2, ... , *n*}, choose *zi* ∈ *B*(*xi*,*ε*) ∩ *Y* and notice that *zi* ∈/ *A* ∪ *L* ∪ [*rj*, *pj*] if *i* = *j*, and hence *zi* ∈ (*ri*, *pi*). Now, if *x* ∈ *A*, there exists *zi*, *zj* which are in different components of *<sup>K</sup>* − {*x*}. Then, *<sup>x</sup>* <sup>∈</sup> [*zi*, *zj*], which shows that *<sup>A</sup>* <sup>⊆</sup> *i*,*j* [*zi*, *zj*]; since *Y*

is arcwise connected, *i*,*j* [*zi*, *zj*] ⊆ *Y* and therefore *A* ⊆ *Y*; in particular, no point belonging

to *A* is a terminal vertex of *Y*.

We want to see that *Y* contains exactly *n* terminal vertices and these are contained in the arcs (*ri*, *pi*). Let *y* ∈ *Y* be a terminal vertex of *Y*. Since *y* ∈/ *L*, we have *y* = *pi* for all 1 ≤ *i* ≤ *n* and *y* ∈/ *A* gives *y* ∈ (*ri*, *pi*) for some *i*. For the above argument, it follows that *Y* contains at most *n* terminal vertices; otherwise, two of them must belong to a same arc (*ri*, *pi*) which is not possible.

Now, given 1 ≤ *i* ≤ *n*, *Gi* = *Y* ∪ [*ri*, *pi*] is a subspace of *G*. Since *G* is hereditary unicoherent, *Y* ∩ [*ri*, *pi*] is connected and non-degenerate (i.e., contains more than one point) because the arc [*ri*, *zi*] is contained in the intersection and therefore such intersection is an arc whose extremes are *ri* and say *yi*. The point *yi* is a terminal vertex of *Y*. This shows that *Y* contains at least *n* terminal vertices. We conclude *Y* = *C***<sup>y</sup>** ∈ (M(*A*))◦, where *y* = (*yi*)*<sup>n</sup> <sup>i</sup>*=1.

Let us verify that *n*-cell M(*A*) is actually maximal; for this purpose, suppose there exists a free *n*-cell A ⊆ *C*(*G*) such that M(*A*) ⊆ A with M(*A*) = A. By Corollary 1, it must occur that there exists some point *C***<sup>x</sup>** ∈ *∂*M(*A*) such that *C***<sup>x</sup>** ∈ A◦. Take *ε* > 0 such that *B*(*C***x**,*ε*) ⊆ A◦.

Now, there are several cases to consider about the point *C***x**. The first one arises when we suppose *C***<sup>x</sup>** = *A* ∪ *<sup>n</sup> i*=1 [*ri*, *xi*] , where, for some index, say *i* = 1, we have *x*<sup>1</sup> = *p*<sup>1</sup> is a

terminal vertex of *K* and, at the same time, a ramification vertex of the dendrite *G*.

Let *ui* ∈ [*ri*, *xi*] (for 2 ≤ *i* ≤ *n*) be points such that [*ui*, *xi*] ⊆ *B*(*xi*,*ε*) and let *L*1, *Ln*+<sup>1</sup> be two different edges of *G* such that *L*<sup>1</sup> ∩ *Ln*+<sup>1</sup> ∩ *K* = {*p*1}. Consider also points *u*<sup>1</sup> ∈ *L*<sup>1</sup> and *un*+<sup>1</sup> ∈ *Ln*+<sup>1</sup> such that [*u*1, *p*1] ⊆ *B*(*p*1,*ε*) and [*un*+1, *p*1] ⊆ *B*(*p*1,*ε*).

For *y*<sup>1</sup> ∈ [*p*1, *u*1], *yi* ∈ [*ui*, *xi*] (2 ≤ *i* ≤ *n*) and *yn*+<sup>1</sup> ∈ [*p*1, *un*+1], let *A*<sup>1</sup> = [*p*1, *y*1], *Ai* = [*ui*, *yi*] and *An*+<sup>1</sup> = [*pi*, *yn*+1]. The family of all subspaces of the form *A* ∪ *n* +1 *i*=1 *Ai* is an

(*n* + 1)-cell contained in *B*(*C***x**,*ε*) ⊆ A◦, and this is a contradiction. Similar considerations show that, if *xi* is an essential point for some *i*, then it is possible find an (*n* + 1)-cell contained in A◦.

A second case is obtained when, for some index *i*, say *i* = 1, it occurs that *x*<sup>1</sup> = *r*1. In this case, *C***<sup>x</sup>** can not belong to Int (M(*A*)) since this contradicts Lemma 5. However,

$$\text{if } \mathbb{C}\_{\mathbf{x}} \in \text{Fr}(\mathfrak{M}(A)), \text{ consider the decomposition } \mathbb{C}(K) = \left( \bigcup\_{T \in IT(K)} \mathfrak{M}(T) \right) \cup \left( \bigcup\_{I \in E(K)} \mathbb{C}(I) \right)$$

$$\text{of Lemma 7 (ii). We claim that we may suppose } \mathbb{C}\_{\mathbf{x}} \in \left( \bigcup\_{T \in \mathbf{x}} \mathfrak{M}(T) \right) \cup \left( \bigcup\_{I \in \mathbf{x}} \mathbb{C}(I) \right).$$

*T*=*A I*∈*E*(*K*) where *T* runs over all internal trees of *K* different from *A*. Otherwise, there exists an open

$$\text{1 set } \mathcal{U} \text{ of } \mathcal{C}(G) \text{ such that } \mathsf{C}\_{\mathbf{x}} \in \mathcal{U} \subseteq \mathcal{C}(G) - \left( \bigcup\_{T \neq A} \mathfrak{M}(T) \right) \cup \left( \bigcup\_{I \in E(K)} \mathcal{C}(I) \right).$$

Let *N* be the first positive integer such that *B*(*C***x**, <sup>1</sup> *<sup>N</sup>* ) ⊆ A◦ ∩ U. Thus, for each *<sup>m</sup>* <sup>≥</sup> *<sup>N</sup>*, there exists *Ym* <sup>∈</sup> *<sup>C</sup>*(*G*) <sup>−</sup> *<sup>C</sup>*(*K*), such that *<sup>H</sup>*(*Ym*, *<sup>C</sup>***x**) <sup>&</sup>lt; <sup>1</sup> *<sup>m</sup>* . For each *m* ≥ *N*, take a point *ym* <sup>∈</sup> *Ym* <sup>−</sup> *<sup>K</sup>* and a point *xm* <sup>∈</sup> *<sup>C</sup>***<sup>x</sup>** such that *<sup>d</sup>*(*ym*, *xm*) <sup>&</sup>lt; <sup>1</sup> *<sup>m</sup>* . Since *G* is compact, the sequence (*ym*) contains a convergent subsequence. We can suppose without loss of generality that (*ym*) is actually convergent, say, to *y*. We claim that *y* ∈ *C***x**. Indeed, given *<sup>ε</sup>* <sup>&</sup>gt; 0, choose *<sup>M</sup>* <sup>∈</sup> <sup>N</sup> such that *ym* <sup>∈</sup> *<sup>B</sup>*(*y*, *<sup>ε</sup>* <sup>2</sup> ) for all *<sup>m</sup>* <sup>≥</sup> *<sup>M</sup>*. If *<sup>m</sup>* <sup>≥</sup> *<sup>M</sup>* satisfies <sup>1</sup> *<sup>m</sup>* <sup>&</sup>lt; *<sup>ε</sup>* <sup>2</sup> , then *<sup>d</sup>*(*xm*, *<sup>y</sup>*) <sup>≤</sup> *<sup>d</sup>*(*xm*, *ym*) + *<sup>d</sup>*(*ym*, *<sup>y</sup>*) <sup>&</sup>lt; <sup>1</sup> *<sup>m</sup>* <sup>+</sup> *<sup>ε</sup>* <sup>2</sup> < *ε*, that is, *xm* ∈ *C***<sup>x</sup>** ∩ *B*(*y*,*ε*). With *ε* > 0 being arbitrary, we conclude that *y* ∈ *C***x**.

The above argument shows that *y* is a cluster point of *G* − *K*; this implies that *y* = *pi* for some index *i*, where *pi* ∈ *R*(*G*) ∪ *ES*(*G*), and this case has already been analyzed.

Thus, we may suppose *C***<sup>x</sup>** ∈ ⎛ <sup>⎝</sup> *T*=*A* M(*T*) ⎞ ⎠ ∪ ⎛ <sup>⎝</sup> *I*∈*E*(*K*) *C*(*I*) ⎞ ⎠. In fact, by [10], [6.2, 6.3] or [12], [Lemma 2.6], we must suppose *C***<sup>x</sup>** ∈ ⎛ <sup>⎝</sup> *T*=*A ∂*M(*T*) ⎞ ⎠ ∪ ⎛ <sup>⎝</sup> *I*∈*E*(*K*) *∂C*(*I*) ⎞ ⎠. Supposing first that *C***<sup>x</sup>** ∈ *∂C*(*I*) since the points belonging to the boundary manifold of a cell are cluster points of their interior manifold, we must have A◦ ∩ (*C*(*I*))◦ = ∅. Now, this set is open in A, and, on the other hand, is contained in *C*(*I*); this is impossible since *Dim*(A) = *n* > 2 = *Dim*(*C*(*I*)).

Suppose now *C***<sup>x</sup>** ∈ *∂*M(*T*). Recall that *A* is the internal tree obtained from *K* by removing their terminal edges. It follows by [10], [5.3, 7.1] that *Dim*(M(*T*)) < *Dim*(M(*A*)). On the other hand, since *C***<sup>x</sup>** is a cluster point of (M(*T*))◦, it must occur that (A)◦ ∩ (M(*T*))◦ = ∅ and notice that this set is open in *C*(*G*) and therefore open in A◦. Hence, there exists a homeomorphic copy of [0, 1] *<sup>n</sup>* contained in (M(*T*))◦, which is impossible regarding the dimension of M(*T*).

The final case to consider is obtained when, for some index *i*, *xi* = *pi* with *pi* ∈ *T*(*G*) or else *xi* ∈ (*ri*, *pi*). In this case, it is not difficult see that *C***<sup>x</sup>** ∈ Int (M(*T*)) contradicting Lemma 5. This shows that M(*A*) is a maximal free *n*-cell as desired.

Conversely, let A be a free *n*-cell, *B* ∈ A◦ and let us analyze how *B* looks. Let *T*(*B*) = {*p*1, ... , *pk*} and let *r*1, ... ,*rs* ∈ *B* − *T*(*B*) be the points such that *oB*(*ri*) < *oG*(*ri*). Put *α<sup>i</sup>* = *oG*(*ri*) − *oB*(*ri*) and assume that *k* + *s* ∑ *i*=1 *α<sup>i</sup>* = *m* > *n*. Consider *ε* > 0 such that *<sup>B</sup>*(*B*,*ε*) ⊆ A◦ and for each 1 ≤ *<sup>i</sup>* ≤ *<sup>s</sup>* consider also arcs [*ui*<sup>1</sup> ,*ri*], ... , [*ui<sup>α</sup><sup>i</sup>* ,*ri*] such that [*uij* ,*ri*] ⊆ *B*(*ri*,*ε*) and [*uij* ,*ri*] ∩ *B* = {*ri*}. In addition, take points *vt* on the terminal edges of *B* such that [*vt*, *pt*] ⊆ *B*(*pt*,*ε*) for 1 ≤ *t* ≤ *k*.

Letting *B*<sup>1</sup> = *B* − *<sup>n</sup> i*=1 [*pt*, *vt*] , we obtain that the family H of all subspaces of *G* has

the form:

$$B\_1 \cup \left(\bigcup\_{t=1}^k [v\_{t\nu} \ge\_t] \right) \cap \left(\bigcup\_{i=1}^s \left(\bigcup\_{j=1}^{\alpha\_i} [r\_{i\nu} y\_{i\_j}] \right) \right) \nu$$

where *xt* ∈ [*vt*, *pt*] and *yij* ∈ [*ri*, *uij* ] is a *m*-cell contained in A, which is absurd. Notice that the above argument in particular shows that *B* − *T*(*B*) does not contain *I*-essential points. A similar reasoning shows that *B* − *T*(*B*) also does not contain *I I*-essential points. Now, assume that *m* < *n*. If *p*1, *p*2, ... , *pq* are the terminal vertices of *B* which are ordinal points of *G*, for each 1 ≤ *t* ≤ *q*, let *Jt* be the edge of *G* such that *pt* ∈ *Jt* and for each *<sup>i</sup>* ∈ {1, ... ,*s*}, let *Ii*<sup>1</sup> , *Ii*<sup>2</sup> , ... , *Ii<sup>α</sup><sup>i</sup>* be the edges of *<sup>G</sup>* such that *Jij* ∩ *<sup>B</sup>* = {*ri*}. Hence, the tree

*K* = *B* ∪ *q t*=1 *Jt* ∪ ⎛ ⎝*s i*=1 ⎛ ⎝*αi j*=1 *Iij* ⎞ ⎠ ⎞ ⎠ has *m* terminal points and satisfies conditions (i) and

(ii) and, by the *only if* part, we have already seen how to get a maximal free *m*-cell containing the above *m*-cell H. Now, on the one hand, by Lemma 1, the cell H is free; on the other hand, since H⊆A◦, Lemma 1 (a) gives that H is not a free cell and, again, this is absurd. Thus, we conclude that *m* = *n* and *K* is the desired tree.

#### **5. Free Cells in Hyperspace of Dendroids**

In this section, necessary and sufficient conditions are given so that Theorem 2 can be applied for dendroids. For this purpose, the notion of convergence space is required.

A non degenerated subspace *A* of a space *X* is called *convergence space* if there exists a sequence *An* of subspaces of *X* such that:


The subspaces *An* can be chosen to be mutually disjoint (see [13] [5.11]).

**Theorem 3.** *Let G be a dendroid, a tree K* ⊆ *G, which satisfies the following conditions:*


*If A is the tree obtained from K by removing their terminal edges, then A induces a maximal free n-cell* M(*A*) *if and only if this cell does not contain convergence subspaces.*

**Proof.** The cell M(*A*) is constructed as in the proof of Theorem 2. It is not hard to see that, if *C***<sup>x</sup>** ∈ (M(*A*))◦ is a convergence subspace, then M(*A*) can not be a free *n*-cell. On the other hand, if M(*A*) is not a free *n*-cell, then there exists *Y* = *C***<sup>x</sup>** ∈ (M(*A*))◦ such that, for each *ε* > 0, there exists *Z* ∈ *C*(*G*) − (M(*A*))◦ with *H*(*Y*, *Z*) < *ε*.

Consider *α<sup>i</sup>* = *d*(*Y*, *pi*), *β* = *H*(*Y*, *∂*(M(*A*))), *γT* = *H*(*Y*,(M(*A*))) and *δ<sup>I</sup>* = *H*(*Y*, *C*(*I*)) (where *T* runs over the set of internal trees of K with *T* = *A* and *I* runs over the set of edges of *K*). Since all these quantities are positive, take *ε* > 0 less than all of them and take *Z*<sup>1</sup> ∈ *C*(*X*) − (M(*A*)) such that *H*(*Z*1,*Y*) < *ε*. If *Z*<sup>1</sup> ∩ *Y* = ∅, we have the following cases:


$$\text{in this case, } Z\_1 \in \mathcal{C}(K) = \left[ \bigcup\_{T \in IT(K)} \mathfrak{M}(T) \right] \cup \left[ \bigcup\_{I \in E(K)} \mathcal{C}(I) \right].$$

If *Z*<sup>1</sup> ∈ *∂*M(*A*), *Z*<sup>1</sup> ∈ M(*T*) with *T* = *A* or *Z*<sup>1</sup> ∈ *C*(*I*), again this contradicts the choice of *ε*1. Therefore, *Z*<sup>1</sup> and *Y* are disjoint. Taking 0 < *ε*<sup>2</sup> < *H*(*Z*1,*Y*), in a similar way, we can obtain a subspace *Z*<sup>2</sup> with no points in common with *Y* and such that *H*(*Z*2,*Y*) < *ε*2. Continuing with this process, we obtain a sequence (*Zn*) of mutually disjoint subspaces convergent to *Y*.

#### **6. Characterization of the Arc in Terms of Anchored Hyperspaces**

The aim of this section is to prove that the arc is the unique arcwise connected space *X*, for which *Cp*(*X*) is an arc for some *p* ∈ *X* (Theorem 4). An important tool in the proof of this theorem is the use of order arcs. An *order arc* in 2*<sup>X</sup>* is an arc *α* contained in 2*<sup>X</sup>* such that, for any *A*, *B* ∈ *α*, *A* ⊆ *B* or *B* ⊆ *A*. The concepts and results we use for order arcs can be found in [13]. We use freely the notation found in there.

#### **Proposition 1.** *The anchored hyperspace Cp*(*X*) *is an arc if and only if it is an order arc.*

**Proof.** Let *α* an order arc in *C*(*X*) from {*p*} to *X*. Since *p* ∈ *A* for all *A* ∈ *α*, we have *α* ⊆ *Cp*(*CX*). Now, it is sufficient to show that {*p*} and *X* are also the end points of *Cp*(*X*), and this will be done by proving that neither {*p*} nor *X* are cut points of *Cp*(*X*) (see [16], [Theorem 1, Pag. 179]). Take different points *A*, *B* ∈ *Cp*(*X*) − {*p*} if *β* and *γ* are order arcs from *A* to *A* ∪ *B* and from *B* to *A* ∪ *B* respectively, then *β* ∪ *γ* ⊆ *Cp*(*X*) − {*p*} is an arc containing the points *A* and *B*; this shows that {*p*} is not a cut point of *Cp*(*X*). Similarly, if *A*, *B* ∈ *Cp*(*X*) − {*X*}, taking *β* and *γ* order arcs from {*p*} to *A* and from {*p*} to *B*, one obtains that *X* is not a cut point either and therefore *α* = *Cp*(*X*). The converse is obvious.

A point *p* of a space *X* is an *irreducibility point* of *X* if there exists another point *q* such that no proper subspace contains both points. The following result is due to Kuratoski and is a handy tool in the proof of Theorem 4.

**Lemma 8** (Kuratoski's Theorem, [15])**.** *Let X be a space and let p* ∈ *X. Then, p is point of irreducibility of X if and only if X is not the union of two proper subspaces both of which contain p.*

**Theorem 4.** *Let X be an arcwise connected space. Then, Cp*(*X*) *is an arc for some p* ∈ *X if and only if X is an arc.*

**Proof.** By Proposition 1, *Cp*(*X*) is an order arc from {*p*} to *X*. It follows that *X* is not the union of two proper subspaces both containing the point *p*. By Lemma 8, it turns out that *p* is an irreducibility point of *X*, if *q* ∈ *X* is another point such that no proper subspace of *X* contains the points *p* and *q*; the arcwise connectedness implies that *X* = [*p*, *q*].

For the converse, suppose without loss of generality that *X* = [0, 1]. Letting *p* = 0, the map

*x* → [*p*, *x*],

is a homeomorphism from *X* to *Cp*(*X*).

The arcwise connectedness hypothesis is necessary in the above theorem (see [13] [Example 1.1]).

#### **7. Comparative Studies and Conclusions**

Some of the main goals on hyperspace research from a theoretical approach are: to obtain topological models corresponding to familiar or not difficult to handle spaces, to find relations between hyperspaces and their underlying spaces, uniqueness of hyperspaces, i.e., to investigate which spaces are the only ones whose hyperspaces possess a given structure. Motivated by the studies carried out in [17,18], the present work was deemed convenient by the authors. In the aforementioned works, the existence of cells in hyperspaces is characterized. Our work is carried out on infinite graphs and describes when such cells are free.

In [19], the arc is characterized in terms of anchored hyperspaces within the class of trees. In our work, we conduct a similar study but within a broader class of spaces, the arc-connected spaces.

Question: If the class of anchored hyperspaces of an arcwise connected space *X* matches the class of anchored hyperspaces of a connected graph *G*, does it follow that *X* and *G* are homeomorphic?

**Author Contributions:** Writing—original draft, J.Á.J.M., G.R.H., J.R.V. and O.R.C. All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This article was supported by the Perfil Deseable del Programa para el Desarrollo Profesional Docente (PRODEP), a federal institution of México's government and the Universidad Autónoma de Guerrero (UAGro).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

