**1. Introduction**

The issue of reducing the number of crossings on edges of simple graphs is interesting in a lot of areas. Probably one of the most popular areas is the implementation of the VLSI layout because it caused a significant revolution in circuit design and thus had a strong effect on parallel calculations. Crossing numbers have also been studied to improve the readability of hierarchical structures and automated graphs. The visualized graph should be easy to read and understand. For the sake of clarity of graphic drawings, some reduction of an edge crossing is probably the most important. Note that examining number of crossings of simple graphs is an NP-complete problem by Garey and Johnson [1].

The crossing number cr(*G*) of a simple graph *G* with the vertex set *V*(*G*) and the edge set *E*(*G*) is the minimum possible number of edge crossings in a drawing of *G* in the plane (for the definition of a drawing see Klešˇc [2]). One can easily verify that a drawing with the minimum number of crossings (an optimal drawing) is always a good drawing, meaning that no two edges cross more than once, no edge crosses itself, and also no two edges incident with the same vertex cross. Let *D* be a good drawing of the graph *G*. We denote the number of crossings in *D* by cr*D*(*G*). Let *Gi* and *Gj* be edge-disjoint subgraphs of *G*. We denote the number of crossings between edges of *Gi* and edges of *Gj* by cr*D*(*Gi*, *Gj*), and the number of crossings among edges of *Gi* in *D* by cr*D*(*Gi*). For any three mutually edge-disjoint subgraphs *Gi*, *Gj*, and *Gk* of *G* by [2], the following equations hold:

$$\text{crr}\_D(\mathbf{G}\_i \cup \mathbf{G}\_j) = \text{crr}\_D(\mathbf{G}\_i) + \text{crr}\_D(\mathbf{G}\_j) + \text{crr}\_D(\mathbf{G}\_{i\prime} \mathbf{G}\_j) \; ,$$

$$\text{crr}\_D(\mathbf{G}\_i \cup \mathbf{G}\_{j\prime} \mathbf{G}\_k) = \text{crr}\_D(\mathbf{G}\_{i\prime} \mathbf{G}\_k) + \text{cr}\_D(\mathbf{G}\_{j\prime} \mathbf{G}\_k) \; .$$

Throughout this paper, some parts of proofs will be based on Kleitman's result [3] on crossing numbers for some complete bipartite graphs *Km*,*<sup>n</sup>* on *m* + *n* vertices with a partition *V*(*Km*,*n*) = *V*<sup>1</sup> ∪ *V*<sup>2</sup> and *V*<sup>1</sup> ∩ *V*<sup>2</sup> = ∅ containing an edge between every pair of vertices from *V*<sup>1</sup> and *V*<sup>2</sup> of sizes *m* and *n*, respectively. He showed that

$$\text{cr}(K\_{m,n}) = \left\lfloor \frac{m}{2} \right\rfloor \left\lfloor \frac{m-1}{2} \right\rfloor \left\lfloor \frac{n}{2} \right\rfloor \left\lfloor \frac{n-1}{2} \right\rfloor, \quad \text{if} \quad \min\{m, n\} \le 6. \tag{1}$$

**Citation:** Staš, M. Parity Properties of Configurations. *Mathematics* **2022**, *10*, 1998. https://doi.org/10.3390/ math10121998

Academic Editors: Irina Cristea and Hashem Bordbar

Received: 15 May 2022 Accepted: 7 June 2022 Published: 9 June 2022

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For an overview of several exact values of crossing numbers for specific graphs or some families of graphs, see Clancy [4]. The main goal of this survey is to summarize all such published results for crossing numbers along with references also in an effort to give priority to the author who published the first result. Chapter 4 is devoted to the issue of crossing numbers of join product with all simple graphs of order at most six mainly due to unknown values of cr(*Km*,*n*) for both *m*, *n* more than six in (1). The join product of two graphs *Gi* and *Gj*, denoted *Gi* + *Gj*, is obtained from vertex-disjoint copies of *Gi* and *Gj* by adding all edges between *V*(*Gi*) and *V*(*Gj*). For |*V*(*Gi*)| = *m* and |*V*(*Gj*)| = *n*, the edge set of *Gi* + *Gj* is the union of the disjoint edge sets of the graphs *Gi*, *Gj*, and the complete bipartite graph *Km*,*n*. Let *Dn* denote the discrete graph (sometimes called empty graph) on *n* vertices, and let *Kn* be the complete graph on *n* vertices. The exact values for crossing numbers of *G* + *Dn* for all graphs *G* of order at most four are given by Klešˇc and Schrötter [5], and also for a lot of connected graphs *G* of order five and six [2,6–24]. Note that cr(*G* + *Dn*) are known only for some disconnected graphs *G*, and so the purpose of this paper is to extend known results concerning this topic to new disconnected graphs [25–28].

Let *G*<sup>∗</sup> = (*V*(*G*∗), *E*(*G*∗)) be the disconnected graph of order five consisting of two components isomorphic to the complete graphs *K*<sup>2</sup> and *K*3, respectively, and let also *V*(*G*∗) = {*v*1, *v*2, ... , *v*5}. We cannot determine the crossing number of the join product *G*<sup>∗</sup> + *Dn* by a similar technique like in [2,18] because |*E*(*G*∗)| < |*V*(*G*∗)|. From the topological point of view, number of crossings of any drawing *D* of *G*<sup>∗</sup> + *Dn* placed on surface of the sphere does not matter which of regions is unbounded, but on how many times edges of the graph *G*<sup>∗</sup> could be crossed by a subgraph *T<sup>i</sup>* in *D*. This representation of *T<sup>i</sup>* best describes idea of a configuration utilizing some cyclic permutation on pre-numbered vertices of *G*∗.

**Theorem 1.** cr(*G*<sup>∗</sup> + *D*1) = 0 *and* cr(*G*<sup>∗</sup> + *Dn*) = *n*<sup>2</sup> − 2*n* + - *<sup>n</sup>* 2 . *for n* ≥ 2*, i.e.,* cr(*G*<sup>∗</sup> + *Dn*) = 4 - *n* 2 .- *<sup>n</sup>*−<sup>1</sup> 2 . + - *<sup>n</sup>* 2 . *for n even and* cr(*G*<sup>∗</sup> + *Dn*) = 4 - *n* 2 .- *<sup>n</sup>*−<sup>1</sup> 2 . + - *<sup>n</sup>* 2 . − 1 *for n odd at least 3.*

All subcases of the proof of Theorem 2 will be clearer if a graph of configurations G*<sup>D</sup>* is used as a graphical representation of minimum numbers of crossings between two different subgraphs. Moreover, in the case of our symmetric graph *G*∗, the graph G*<sup>D</sup>* can be linked to parity properties of configurations. Our proof of the main Theorem 2 is therefore an inevitable combination of topological analysis of existing configurations with their parity properties. The color resolution of weighted edges in G*<sup>D</sup>* will also serve us for a simpler description of existence of its possible subgraphs in the examined drawing *D* of *G*<sup>∗</sup> + *Dn*. Software COGA [29] should be also very helpful in certain parts of presented proofs mainly due to possibility of generating all cyclic permutations of five elements and counting of their subsequent interchanges of adjacent elements.

The obtained crossing number of the join product *G*<sup>∗</sup> + *Dn* is in very special form which is caused by a completely different behavior for *n* even and odd number. The paper concludes by giving crossing numbers of *G*<sup>∗</sup> + *Pn* and *G*<sup>∗</sup> + *Cn* with same values in Corollaries 3 and 4, respectively, that is something unique in the crossing number theory.

#### **2. Cyclic Permutations and Corresponding Configurations**

The join product *G*<sup>∗</sup> + *Dn* (sometimes used notation *G*<sup>∗</sup> + *nK*1) consists of one copy of the graph *G*<sup>∗</sup> and *n* vertices *t*1, ... , *tn*, and any vertex *ti* is adjacent to every vertex of the graph *G*∗. We denote the subgraph induced by five edges incident with the fixed vertex *ti* by *T<sup>i</sup>* , which yields that

$$G^\* + D\_n = G^\* \cup K\_{5,n} = G^\* \cup \left(\bigcup\_{i=1}^n T^i\right). \tag{2}$$

We consider a good drawing *D* of *G*<sup>∗</sup> + *Dn*. By the rotation rot*D*(*ti*) of a vertex *ti* in *D* we understand the cyclic permutation that records the (cyclic) counterclockwise order in which edges leave *ti*, as defined by Hernández-Vélez et al. [30] or Woodall [31]. We use the notation (12345) if the counter-clockwise order of edges incident with the fixed vertex *ti* is *tiv*1, *tiv*2, *tiv*3, *tiv*4, and *tiv*5. We recall that rotation is a cyclic permutation. By rot*D*(*ti*), we understand the inverse permutation of rot*D*(*ti*). In the given drawing *D*, it is highly desirable to separate *n* subgraphs *T<sup>i</sup>* into three mutually disjoint subsets depending on how many times edges of *G*<sup>∗</sup> could be crossed by *T<sup>i</sup>* in *D*. Let us denote by *RD* and *SD* the set of subgraphs for which cr*D*(*G*∗, *T<sup>i</sup>* ) = 0 and cr*D*(*G*∗, *T<sup>i</sup>* ) = 1, respectively. Edges of *G*<sup>∗</sup> are crossed by each remaining subgraph *T<sup>i</sup>* at least twice in *D*.

First, note that if *D* is a drawing of the join product *G*<sup>∗</sup> + *Dn* with the empty set *RD*, then ∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> cr*D*(*G*∗, *<sup>T</sup><sup>i</sup>* ) ≥ *n* enforces at least *n*<sup>2</sup> − 2*n* + - *<sup>n</sup>* 2 . crossings in *D* provided by

$$\text{crr}\_D(\mathcal{G}^\* + D\_n) \ge \text{cr}\_D(\mathcal{K}\_{5,n}) + \text{cr}\_D(\mathcal{G}^\*, \mathcal{K}\_{5,n}) \ge 4\left\lfloor \frac{n}{2} \right\rfloor \left\lfloor \frac{n-1}{2} \right\rfloor + n \ge n^2 - 2n + \left\lfloor \frac{n}{2} \right\rfloor.$$

Based on this argument, we will only consider drawings of the graph *G*<sup>∗</sup> for which there is a possibility to obtain a subgraph *T<sup>i</sup>* ∈ *RD*. Moreover, let *F<sup>i</sup>* denote the subgraph *G*<sup>∗</sup> ∪ *T<sup>i</sup>* for any *T<sup>i</sup>* ∈ *RD*, which yields that each such subgraph *F<sup>i</sup>* is represented by its rot*D*(*ti*).

Let us discuss all possible subdrawings of *G*<sup>∗</sup> induced by *D*. As edges of its subgraph isomorphic to *K*<sup>3</sup> do not cross each other, it is obvious there are only two such possible drawings of *G*<sup>∗</sup> presented in Figure 1.

**Figure 1.** Two possible non isomorphic drawings of the graph *G*∗. (**a**): the planar drawing of *G*∗; (**b**): the drawing of *G*<sup>∗</sup> with two crossings among edges.

Assume there is a good drawing *D* of *G*<sup>∗</sup> + *Dn* with planar subdrawing of the graph *G*<sup>∗</sup> induced by *D* and also the vertex notation of *G*<sup>∗</sup> in such a way as shown in Figure 1a. Our aim is to list all possible rotations rot*D*(*ti*) which can appear in *D* if edges of *G*<sup>∗</sup> are not crossed by *T<sup>i</sup>* . Since there is only one subdrawing of *F<sup>i</sup>* \ {*v*4, *v*5} represented by the rotation (132), there are three possibilities to obtain the subdrawing of *F<sup>i</sup>* without the edge *v*4*v*<sup>5</sup> depending on in which region both edges *tiv*<sup>4</sup> and *tiv*<sup>5</sup> are placed. Of course, there are two next ways how to place the corresponding two edges together with the edge *v*4*v*<sup>5</sup> for each mentioned case. These 3 × 2 = 6 possibilities under our consideration can be denoted by A*k*, for *k* = 1, ... , 6. We will call them by the *configurations* of corresponding subdrawings of the subgraph *G*<sup>∗</sup> ∪ *T<sup>i</sup>* in *D* and suppose their drawings as shown in Figure 2.

In the rest of the paper, we present a cyclic permutation by the permutation with 1 in the first position. Thus, the configurations A1, A2, A3, A4, A5, and A<sup>6</sup> are represented by the cyclic permutations (13245), (13254), (14532), (15432), (13452), and (13542), respectively. Clearly, in a fixed drawing of the graph *G*<sup>∗</sup> + *Dn*, some configurations from M = {A1, A2, A3, A4, A5, A6} need not appear. We denote by M*<sup>D</sup>* the set of all configurations that exist in the drawing *D* belonging to the set M.

**Figure 2.** Drawings of six possible configurations A*<sup>k</sup>* of subgraph *F<sup>i</sup>* = *G*<sup>∗</sup> ∪ *T<sup>i</sup>* for *T<sup>i</sup>* ∈ *RD*.

Let X , Y be two configurations from M*<sup>D</sup>* (not necessary distinct). We denote the number of edge crossings between two different subgraphs *T<sup>i</sup>* and *T<sup>j</sup>* with conf(*F<sup>i</sup>* ) = X and conf(*F<sup>j</sup>* ) = Y in *D* by cr*D*(X , Y). Finally, let cr(X , Y) = min{cr*D*(X , Y)} among all good drawings of *G*<sup>∗</sup> + *Dn* with the planar subdrawing of *G*<sup>∗</sup> induced by *D* given in Figure 1a and with X , Y∈M*D*. Our aim shall be to establish cr(X , Y) for all pairs X , Y∈M. In particular, the configurations A<sup>1</sup> and A<sup>4</sup> are represented by the cyclic permutations (13245) and (15432), respectively. Each subgraph *T<sup>j</sup>* with conf(*F<sup>j</sup>* ) = A<sup>4</sup> crosses edges of each *T<sup>i</sup>* with conf(*F<sup>i</sup>* ) = A<sup>1</sup> at least once provided by the minimum number of interchanges of adjacent elements of (13245) required to produce (15432)=(12345) is one, i.e., cr(A1, A4) ≥ 1. For more details see also Woodall [31]. The same reason gives cr(A1, A2) ≥ 3, cr(A1, A3) ≥ 2, cr(A1, A5) ≥ 2, cr(A1, A6) ≥ 1, cr(A2, A3) ≥ 1, cr(A2, A4) ≥ 2, cr(A2, A5) ≥ 1, cr(A2, A6) ≥ 2, cr(A3, A4) ≥ 3, cr(A3, A5) ≥ 2, cr(A3, A6) ≥ 1, cr(A4, A5) ≥ 1, cr(A4, A6) ≥ 2, and cr(A5, A6) ≥ 3. Clearly, also cr(A*k*, A*k*) ≥ 4 for any *k* = 1, ... , 6. The lower bounds obtained for number of crossings between two configurations from M are summarized in the symmetric Table 1 (here, conf(*F<sup>i</sup>* ) = A*<sup>k</sup>* and conf(*F<sup>j</sup>* ) = A*<sup>l</sup>* with *k*, *l* ∈ {1, ... , 6}). Note that these values cannot be increased, i.e., the lower bounds can be achieved in some subdrawings of *G*<sup>∗</sup> ∪ *T<sup>i</sup>* ∪ *T<sup>j</sup>* for *Ti* , *T<sup>j</sup>* ∈ *RD* with desired configurations.

**Table 1.** The minimum number of crossings between two different subgraphs *T<sup>i</sup>* and *T<sup>j</sup>* such that conf(*F<sup>i</sup>* ) = A*<sup>k</sup>* and conf(*F<sup>j</sup>* ) = A*l*, where the achieved values are color-coded. Namely, the values 1, 2, 3, and 4 will correspond to green, blue, brown, and black, respectively.


Further, due to symmetry of mentioned configurations, let us define two functions

$$
\pi\_1: \{1, 2, 3\} \to \{1, 2, 3\}, \text{ with } \pi\_1(1) = 3, \pi\_1(2) = 1, \text{ and } \pi\_1(3) = 2,
$$

*π*<sup>2</sup> : {4, 5}→{4, 5}, with *π*2(4) = 5, and *π*2(5) = 4.

Let Π1, Π<sup>2</sup> : M→M be the functions obtained by applying *π*<sup>1</sup> and *π*<sup>2</sup> on corresponding cyclic permutations of configurations in M, respectively. Thus, we have

$$
\Pi\_1(\mathcal{A}\_1) = \mathcal{A}\_3,\ \Pi\_1(\mathcal{A}\_3) = \mathcal{A}\_5,\ \Pi\_1(\mathcal{A}\_5) = \mathcal{A}\_1,\ \Pi\_1(\mathcal{A}\_2) = \mathcal{A}\_4.
$$

$$
\Pi\_1(\mathcal{A}\_4) = \mathcal{A}\_{6\prime},\ \Pi\_1(\mathcal{A}\_6) = \mathcal{A}\_{2\prime},\ \Pi\_2(\mathcal{A}\_1) = \mathcal{A}\_{2\prime},\ \Pi\_2(\mathcal{A}\_2) = \mathcal{A}\_{1\prime}.
$$

$$
\Pi\_2(\mathcal{A}\_3) = \mathcal{A}\_4, \Pi\_2(\mathcal{A}\_4) = \mathcal{A}\_3, \Pi\_2(\mathcal{A}\_5) = \mathcal{A}\_6, \Pi\_2(\mathcal{A}\_6) = \mathcal{A}\_5.
$$

Therefore it is not difficult to show that values in rows of Table 1 can be obtained by successive application of the mentioned transformations Π<sup>1</sup> and Π2.

#### **3. The Graph of Configurations and Parity Properties**

Low possible number of crossings between two different subgraphs from the nonempty set *RD* is one of main problems in determining cr(*G*<sup>∗</sup> + *Dn*), and graph of configurations as a graphical representation of Table 1 is going by useful tool in our research. This idea of representation was first introduced in [26].

Let *D* be a good drawing of *G*<sup>∗</sup> + *Dn* with the planar subdrawing of *G*<sup>∗</sup> induced by *D* given in Figure 1a, and let M*<sup>D</sup>* be nonempty set of all configurations that exist in *D* belonging to M = {A1, A2, A3, A4, A5, A6}. A graph of configurations G*<sup>D</sup>* is an ordered triple (*VD*, *ED*, *wD*), where *VD* is the set of vertices, *ED* is the set of edges formed by all unordered pairs of two vertices (not necessary distinct), and a weight function *w* : *ED* → N that associates with each edge of *ED* an unordered pair of two vertices of *VD*. The vertex *ak* ∈ *VD* if the corresponding configuration A*<sup>k</sup>* ∈ M*<sup>D</sup>* for some *k* ∈ {1, ... , 6}. The edge *e* = *ak al* ∈ *ED* if *ak* and *al* are two vertices of G*D*. Finally, *wD*(*e*) = *m* ∈ N for the edge *e* = *ak al*, if *m* is associated lower bound between two configurations A*<sup>k</sup>* and A*<sup>l</sup>* in Table 1. Based on that G*<sup>D</sup>* is an undirected edge-weighted graph without multiple edges uniquely determined by *D* and is also subgraph of G induced by *VD* if we define G = (*V*, *E*, *w*) in the same way over M. The graph G = (*V*, *E*, *w*) corresponds to the edge-weighted complete graph *K*<sup>6</sup> in Figure 3, and thus will follow all subcases in the proof of the main Theorem 2 more clearly. In the rest of Figure 3, let any loop of the mentioned graph G be presented by circle around vertex with respect to weight 4.

**Figure 3.** Representation of lower bounds of Table 1 by the graph G = (*V*, *E*, *w*).

Let *α<sup>i</sup>* denote the number of all subgraphs *T<sup>j</sup>* ∈ *RD* with the configuration A*<sup>i</sup>* ∈ M*<sup>D</sup>* of *F<sup>j</sup>* = *G*<sup>∗</sup> ∪ *T<sup>j</sup>* for each *i* = 1, ... , 6. So, if we denote by *Io* = {1, 3, 5} and *Ie* = {2, 4, 6}, then ∑*i*∈*Io*∪*Ie α<sup>i</sup>* = |*RD*|. Moreover, for a better understanding, we get for all *i* ∈ *Io* ∪ *Ie*: *α<sup>i</sup>* > 0 if and only if there is a subgraph *T<sup>j</sup>* ∈ *RD* with the configuration A*<sup>i</sup>* ∈ M*<sup>D</sup>* of *F<sup>j</sup>* = *G*<sup>∗</sup> ∪ *T<sup>j</sup>* if and only if *ai* ∈ *VD* in the graph G*D*.

Now, let us assume the configurations A<sup>1</sup> of *F<sup>i</sup>* , A<sup>4</sup> of *F<sup>j</sup>* , and A<sup>6</sup> of *Fk*. The reader can easily find a subdrawing of *G*<sup>∗</sup> ∪ *T<sup>i</sup>* ∪ *T<sup>j</sup>* ∪ *T<sup>k</sup>* in which cr*D*(*T<sup>i</sup>* , *T<sup>j</sup>* ) = 1, cr*D*(*T<sup>i</sup>* , *Tk*) = 1, and cr*D*(*T<sup>j</sup>* , *Tk*) = 2, i.e., cr*D*(*T<sup>i</sup>* ∪ *T<sup>j</sup>* ∪ *Tk*) = 4 = cr(*K*5,3). Further, there is a possibility to

add another subgraph *T<sup>l</sup>* that crosses edges of the graph *T<sup>i</sup>* ∪ *T<sup>j</sup>* ∪ *T<sup>k</sup>* four times. We have to emphasize that the vertex *tl* must be placed in the triangular region with three vertices of *G*<sup>∗</sup> on its boundary (in the subdrawing of *G*<sup>∗</sup> ∪ *T<sup>i</sup>* ∪ *T<sup>j</sup>* ∪ *Tk*), i.e., *T<sup>l</sup>* ∈ *RD* ∪ *SD* and the subgraph *F<sup>l</sup>* = *G*<sup>∗</sup> ∪ *T<sup>l</sup>* is represented by rot*D*(*tl*)=(12435). Clearly, the number of adding crossings cannot be smaller than 4 according to the well-known fact that cr(*K*5,4) = 8. This situation suggests one natural problem which requires the following definition of a new number *β*1. If *α*<sup>1</sup> > 0, *α*<sup>4</sup> > 0, and *α*<sup>6</sup> > 0, then let us denote by *β*<sup>1</sup> the number of subgraphs *T<sup>l</sup>* ∈ *RD* ∪ *SD* with rot*D*(*tl*)=(12435). It is obvious that any subgraph *T<sup>l</sup>* ∈ *RD* ∪ *SD* satisfies the condition cr*D*(*G*<sup>∗</sup> ∪ *T<sup>i</sup>* ∪ *T<sup>j</sup>* ∪ *Tk*, *T<sup>l</sup>* ) ≥ 2 + 4 = 6 with the configurations A<sup>1</sup> of *F<sup>i</sup>* , A<sup>4</sup> of *F<sup>j</sup>* , and A<sup>6</sup> of *Fk*, and the number of *T<sup>l</sup>* ∈ *RD* ∪ *SD* that cross the graph *G*<sup>∗</sup> ∪ *T<sup>i</sup>* ∪ *T<sup>j</sup>* ∪ *T<sup>k</sup>* exactly six times is at most *β*1. Due to symmetry of some configurations, it is appropriate to use the transform functions Π1, Π<sup>2</sup> defined above and by the similar way, we can also define the numbers *β<sup>i</sup>* for any *i* = 2, ... , 6. Thus, if *α*<sup>2</sup> > 0, *α*<sup>3</sup> > 0, and *α*<sup>5</sup> > 0 or *α*<sup>3</sup> > 0, *α*<sup>2</sup> > 0, and *α*<sup>6</sup> > 0 or *α*<sup>4</sup> > 0, *α*<sup>1</sup> > 0, and *α*<sup>5</sup> > 0 or *α*<sup>5</sup> > 0, *α*<sup>2</sup> > 0, and *α*<sup>4</sup> > 0 or *α*<sup>6</sup> > 0, *α*<sup>1</sup> > 0, and *α*<sup>3</sup> > 0, then let us denote by *β*<sup>2</sup> or *β*<sup>3</sup> or *β*<sup>4</sup> or *β*<sup>5</sup> or *β*<sup>6</sup> the number of subgraphs *T<sup>l</sup>* ∈ *RD* ∪ *SD* represented by the rotation (12534) or (14253) or (15243) or (15234) or (14235), respectively. The importance of the values *β<sup>i</sup>* will be presented in the proof of the main Theorem 2 as parity properties (6) and (7).
