Bounds on the Clique and the Independence Number for Certain Classes of Graphs

Abstract

In this paper, we study the class of graphs $G_{m,n}$ that have the same degree sequence as two disjoint cliques $K_m$ and $K_n$, as well as the class ${\overline{G}}_{m,n}$ of the complements of such graphs. The problems of finding a maximum clique and a maximum independent set are NP-hard on $G_{m,n}$. Therefore, looking for upper and lower bounds for the clique and independence numbers of such graphs is a challenging task. In this article, we obtain such bounds, as well as other related results. In particular, we consider the class of regular graphs, which are degree-equivalent to arbitrarily many identical cliques, as well as such graphs of bounded degree.

1. Introduction

Two graphs $G_1$ and $G_2$ are called degree-equivalent if they have the same degree sequence. In this paper, we present results about simple graphs that are different from and degree-equivalent to a disjoint union of cliques $K_{p_1},\dots ,K_{p_k}$ (denoted ${\bigcup }_{i=1}^n{K}_{p_i}$), as well as the complements of such graphs. We will denote this class of graphs by $G_{p_1,\cdots ,p_k}$ and the class of their complements by ${\overline{G}}_{p_1,\cdots ,p_k}$. Predominantly, our results hold for graphs that are degree-equivalent to two disjoint cliques and their complements, i.e., the graphs in $G_{m,n}$ and ${\overline{G}}_{m,n}$. Graphs in these families have degree sequence $\{m^{m-1},n^{n-1}\}$ and $\{m^n,n^m\}$, respectively, where $a^x$ means the number a appears x times in the sequence.

The graphs in $G_{m,n}$ and ${\overline{G}}_{m,n}$ may have different applications. For example, they can model peer-to-peer file-sharing networks where, for instance, each of n downloaders must be connected to m peers, and each of m uploaders must be connected to n peers. They can also model networks, which start as disjoint cliques or as complete bipartite graphs and then evolve through a sequence of 2-switchings (see Section 2 for definitions).

The graphs we examine are somewhat reminiscent of biregular, semiregular, almost regular, and nearly regular graphs which have been studied in several works [1,2,3,4,5,6,7,8]. Thus, a graph is biregular [7] (alternatively, semiregular [8]) if it is bipartite for which all vertices of the same part have the same degree. A graph G is nearly regular if $\Delta \left(G\right)-\delta \left(G\right)\le c·\delta \left(G\right)$ [9] (see Section 2 for definitions). A graph G is almost regular if its vertex degrees differ by at most one [2]. See Figure 1 for illustration.

Some variations or generalizations of this definition have been considered as well [10,11]. A special example of a graph that belongs to the class ${\overline{G}}_{m,n}$ is the well-known Mantel graph [12] (also known as the Turán graph $T(n,2)$, see [13,14]). It is a graph on n vertices, which is the complement of a graph that consists of two cliques of size $\lfloor n/2\rfloor$ and $\lceil n/2\rceil$. The graphs of two of the Platonic solids, the cube and the icosahedron, are in the classes $G_{4,4}$ and $G_{6,6}$, respectively. The graph of the dodecahedron is in the class $G_{4,4,4,4}$. When $m=n$, the graphs in $G_{m,n}$ and ${\overline{G}}_{m,n}$ are regular. However, apart from such special cases, not much was known about the graphs in $G_{m,n}$ and ${\overline{G}}_{m,n}$.

Various properties that are shared by all graphs in $G_{m,n}$ and ${\overline{G}}_{m,n}$ were presented in [15]. In particular, it was shown that these graphs can have a rather varied structure yet possess desirable qualities such as Hamiltonicity, traceability, bounded diameter, and efficient recognizability. On the other hand, various optimization problems such as maximum clique, maximum independent set, and minimum vertex cover are NP-hard on these classes of graphs. This last fact suggests looking for efficiently computable upper and lower bounds for the clique number, independence number, and the minimum vertex cover number of such graphs, which is the goal of the present work.

In the next section, we introduce some definitions and known results to be used in the paper. In Section 3 and Section 4, we present, respectively, upper and lower bounds for the clique number and the independence number of graphs in $G_{m,n}$ and ${\overline{G}}_{m,n}$. In Section 5, we present some further results about regular graphs that are degree-equivalent to more than two cliques. We conclude with final remarks and directions for future work in Section 6.

2. Preliminaries

Let $G=(V,E)$ be a simple graph (i.e., with nooops or parallel edges) with vertex set V and edge set E. The neighborhood of a vertex v, denoted $N\left(v\right)$, is the set of vertices adjacent to v; the degree of v, denoted $deg\left(v\right)$, is equal to $\left|N\right(v\left)\right|$, where by $\left|A\right|$ we denote the cardinality of a set A. The degree sequence$D\left(G\right)$ of G is the sequence of its vertex degrees. Graphs $G_1$ and $G_2$ are called degree-equivalent, denoted as $G_1\simeq G_2$, if they have the same degree sequence. $\Delta \left(G\right)$ and $\delta \left(G\right)$, respectively, denote the maximum and the minimum degree of the vertices of a graph G.

G is connected if there is a path that connects any two vertices of G; otherwise, G is disconnected. A (connected) component of G is a maximal connected subgraph of G. G is separable if it is disconnected or can be disconnected by removing a vertex, called a cut-vertex. A graph $G=(U,V,E)$ with a set of vertices $U\cup V$, $U\cap V=\varnothing$, is called bipartite with parts U and V, if every edge of G connects a vertex from U to a vertex from V. A bridge (or cut-edge) of G is an edge whose removal increases the number of components of G. A triangle in G is a cycle with three edges.

Given $S\subset V$, the induced subgraph$G\left[S\right]$ is the subgraph of G whose vertex set is S and whose edge set consists of all edges of G, which have both ends in S. The complement of G is a graph $\overline{G}$ on the same set of vertices such that two vertices of $\overline{G}$ are adjacent if and only if they are not adjacent in G. The distance$d(u,v)$ between vertices u and v in G is the number of edges in a shortest path between u and v in G. The diameter of G is defined as $diam\left(G\right)={max}_{u,v\in V}d(u,v)$.

A clique in graph G is a complete subgraph of G (i.e., a subgraph in which any two vertices are adjacent). A clique on n vertices is denoted by $K_n$. The clique number of G, denoted $\omega \left(G\right)$, is the cardinality of the largest clique in G. An independent (or stable) set of G is a set of vertices, no two of which are adjacent. The independence (or stability) number of G, denoted $\alpha \left(G\right)$, is the cardinality of a largest independent set in G. A vertex cover of G is a set of vertices S such that for every edge of G, at least one of its endpoints is in S. The vertex cover number of G, denoted $\tau \left(G\right)$, is the cardinality of a smallest vertex cover of G. Vertex coloring of G is a coloring of the vertices of G, such that no two adjacent vertices have the same color. The chromatic number of G, denoted $\chi \left(G\right)$, is the smallest number of colors needed to color the vertices of G. Dominating set of G is a subset S of vertices of G, such that any vertex of G is either in S or has a neighbor in S. The domination number of G, denoted $\gamma \left(G\right)$, is the number of vertices in a smallest dominating set of G. Computing $\omega \left(G\right)$, $\alpha \left(G\right)$, $\tau \left(G\right)$, $\chi \left(G\right)$, and $\gamma \left(G\right)$ are classical NP-complete problems [16].

For other graph theoretic definitions and notations, see [17]. Below, we recall several theorems from the literature that we will use in the sequel.

Let $u,v,x,y$ be four vertices in a graph G such that $uv$ and $xy$ are edges of G and $ux$, $vy$, $uy$, and $vx$ are not edges of G. A 2-switching applied to G is an operation that replaces the edges $uv$ and $xy$ with the edges $ux$ and $vy$ or with the edges $uy$ and $vx$. This procedure is illustrated in the following example.

Example 1.

Let $G=K_4\cup K_3$ be the graph in Figure 2, left. Let $uv$ be an edge in $K_4$ and $xy$ be an edge in $K_3$. A two switching of the edges $uv$ and $xy$ leads to one of the graphs in Figure 2, right.

It is well known (and easy to see) that the resulting graph has the same degree sequence as G and that two graphs G and H have the same degree sequence if and only if there is a sequence of 2-switchings that transforms G into H [18]. Moreover, we have the following theorem.

Theorem 1

(Hakimi [19,20]). If $G_1$ and $G_2$ are degree-equivalent graphs, then one can be obtained from the other by a finite sequence of 2-switchings.

Applied to graphs of $G_{p_1\cdots p_k}$, Hakimi’s theorem implies that any such graph can be obtained from several disjoint cliques by a series of 2-switchings.

Hakimi’s theorem makes it clear that any graph H can be “edited” (by replacing some edges with other edges) in such a way that we can obtain any other graph G that is degree-equivalent to H. The total number of edges that are 2-switched can be viewed as edit (or Hamming) distance between graphs G and H. Defining and using distances between graphs appears important in recent studies on large networks when one tries to approximate a large complex graph by a simpler one with similar properties (see [21]). Some of our bounds on clique and independence number of a graph $G\in G_{p_1,p_2}$ are in terms of the number of edges joining vertices of two given cliques $K_{p_1}$ and $K_{p_2}$ from which G can be obtained by a series of 2-switchings. More formally, let $G\simeq K_{p_1}\cup K_{p_2}$. We define a parameter $r=r\left(G\right)$ as the number of edges $uv$ with $deg\left(u\right)=p_1-1$, $deg\left(v\right)=p_2-1$. Thus, the bounds are essentially in terms of the edit distance between G and the two cliques to whose union G is degree-equivalent. Hence, graphs that feature similarly large distances from the same pair of cliques, in turn, are expected to feature similar structural properties.

In obtaining some of our results, we will use several known facts. The following is the earliest result of extremal graph theory.

Theorem 2

(Mantel [12]). The maximum number of edges in a triangle-free graph on n vertices is $\lfloor n^2/4\rfloor$.

The following is a well-known theorem of Caro and Wei providing a lower bound of the independence number of a graph.

Theorem 3

(Caro and Wei [22]). A simple graph G with n vertices and degree sequence $d_1,d_2,\cdots ,d_n$ has an independence number $\alpha \left(G\right)\ge {\sum }_{i=1}^n\frac{1}{d_i+1}$, with equality holding only when G is a disjoint union of n cliques, in which case $\alpha \left(G\right)=n$.

Next, we list a theorem by Erdős, which is a variant of a key result in Ramsey theory [23].

Theorem 4

(Erdős [24]). Define $A\left(n\right)$ as the greatest integer such that given any graph G of n vertices, either it or its complementary graph contains a complete subgraph of order $A\left(n\right)$. Then for $A\left(n\right)\ge 3$, $\frac{logn}{2log2}<A\left(n\right)<\frac{2logn}{log2}$.

We conclude this section by listing some observations and properties of graphs from $G_{m,n}$ (theorem statements along with detailed proofs are available in [15]).

Recall that, by definition, $K_m\cup K_n\notin G_{m,n}$. Thus, there are no graphs in $G_{m,n}$ for $m=1$ or $n=1$, nor for $m=n=2$ since any graph G that is degree-equivalent to $K_m\cup K_n$ for those values of m and n is isomorphic to $K_m\cup K_n$, and therefore $G_{m,n}$ is empty for those values of m and n.

A graph that is degree-equivalent to more than two cliques does not need to be connected (see Figure 3). However, we have the following:

Theorem 5.

Any graph $G\in G_{m,n}$ is connected.

Proof. 

The graph $G=(V,E)$ has $\left|V\right|=m+n$ vertices, of which m of degree $m-1$ and n of degree $n-1$. Assume that G has at least two components. Let $G_1=(V_1,E_1)$ be one of minimum order. If $|V_1|$ < m, then $G_1$ has vertices of degree less than $m-1$, which is a contradiction. If $|V_1|$ > m, then all other components must be of the order less than n. Hence, all vertices of degree $n-1$ must be in $G_1$. Then, $G_1$ has order at least n, which contradicts the minimality of $G_1$. Thus, we obtain that $|G_1|$ = m. Hence, all vertices of $V_1$ must be of degree $m-1$, which is possible only if $G_1=K_m$.

Let $G_2$ be the graph consisting of the other components of G. $G_2$ has n vertices and they are all of degree $n-1$, which is possible only if $G_2=K_n$. Then $G=K_m\cup K_n$, which contradicts $K_m\cup K_n\notin G_{m,n}$. □

We also have the following possibilities regarding cut vertices, bridges, and diameter of graphs in $G_{m,n}$.

Theorem 6.

A graph $G\in G_{m,n}$ must have one of the following:

1. 

No cut-vertices and no bridges

2. 

One cut-vertex and no bridges

3. 

Two cut-vertices and one bridge

4. 

Two cut-vertices and two bridges

5. 

Three cut-vertices and four bridges.

Moreover, if G has at least one cut-vertex, then it is traceable.

Theorem 7.

If $G\in G_{m,n}$, then $diam\left(G\right)\le 4$. If $m=n$, then $diam\left(G\right)\le 3$.

Theorem 8.

If $G\in {\overline{G}}_{m,n}$, then $diam\left(G\right)\le 4$. If $m=n$, then $diam\left(G\right)\le 2$.

3. Bounds on the Clique Number

Regarding possible applications of graphs from the considered classes, information about their clique and independence numbers would be most useful. In this section, we obtain an upper and a lower bound for the clique number of a graph in $G_{m,n}$. We start with a general statement about the possible range of the values of the clique number.

Theorem 9.

Let $G=(V,E)\in G_{m,n}$, $n\ge m\ge 2$. If $m=n=2$, then $\omega \left(G\right)=2$. Otherwise, $2\le \omega \left(G\right)\le n-1$, as the extreme value $n-1$ is reached for some graphs of $G_{m,n}$, for any $m,n\ge 2$ which are not equal to 2 at the same time.

Proof. 

If $m=n=2$, the only graph in $G_{m,n}$ consists of two nonincident edges, as the vertices of each of them make up a 2-clique.

Next, we consider the case where m and n are not both equal to 2. Since G is nonempty, $\omega \left(G\right)\ge 2$. By Mantel’s Theorem [12], if $\left|E\right|>\lfloor n^2/4\rfloor$ then G has a 3-clique, otherwise G may be triangle-free. Therefore, the lower bound 2 may be reached for some values of m and n while may not be reached for others. Next, we will show that $\omega \left(G\right)\le n-1$ and that the upper bound is reached for some graphs of $G_{m,n}$.

Consider first the case $n>m\ge 2$. Start from the graph $K_n\cup K_m$. Let $V_n$ be the set of vertices of $K_n$. Choose two arbitrary vertices $u,v\in V_n$. Switch them with arbitrary two vertices of $K_m$. Let $G_0\in G_{m,n}$ be the obtained graph. Then the complete graphs on $V_n-u$ and $V_n-v$ are $(n-1)$-cliques in $G_0$. Each of its vertices has degree $n-1$ in $G_0$. Clearly, any graph $G^{\prime }$, which is degree-equivalent to G, cannot have a clique larger than $K_n$ since the vertex degrees of $G^{\prime }$ are $n-1$ or $m-1$, and $m-1<n-1$. Since $G^{\prime }$ is connected and has $n+m\ge n+2$ vertices, $G^{\prime }$ cannot have an n-clique, as well, since otherwise, $G^{\prime }$ would have a vertex of degree greater than $n-1$.

Now consider the case $m=n>2$. Clearly, $(n-1)$-cliques can be obtained in the same way as in the case $n>m$ (two in each of the two original cliques $K_n$). One can also obtain a graph $G\in G_{m,n}$ with an $(n-1)$-clique with vertices from each of the two cliques $K_n$, as follows.

Start from the graph $K_n^{\prime }\cup K_n^{″}$, where $K_n^{\prime }$ and $K_n^{″}$ are two copies of $K_n$. First, consider the case where n is even. Choose arbitrary subcliques $K_{n/2}$ of $K_n^{\prime }$ and $K_{n/2-1}$ of $K_n^{″}$. For each vertex v of $K_{n/2}$ and each vertex w of $K_{n/2-1}$, choose $n/2-1$ edges of $K_n^{\prime }$ that are not in $K_{n/2}$ and are incident to v, next choose $n/2$ edges of $K_n^{″}$ that are not in $K_{n/2-1}$ and are incident to w, and then switch each of the chosen edges incident to v with a chosen edge incident to w. Thus, we delete $2(n/2)(n/2-1)$ edges form G and create the same number of new edges; then $(n/2)(n/2-1)$ of the new edges, along with the edges of $K_{n/2-1}$ and $K_{n/2}$, form an $(n-1)$-clique in the obtained graph which belongs to the class $G_{n,n}$.

Now consider the case where n is odd. Choose arbitrary subcliques $K_{\lfloor n/2\rfloor }$ of $K_n^{\prime }$ and $K_{\lfloor n/2\rfloor }$ of $K_n^{″}$ and perform the same series of choosing and switching edges incident to the vertices of $K_n^{\prime }$ and $K_n^{″}$. As a result, we once again obtain an $(n-1)$-clique in the obtained graph, which belongs to $G_{n,n}$.

It is well known that the clique number of a regular graph with k vertices is among the numbers $1,2,3,\cdots ,\lfloor k/2\rfloor ,k$. Clearly, the considered graph $G\in G_{n,n}$ cannot have a clique of size $2n$ (e.g., because its vertices have degree $n-1$). Assume that it can have an n-clique. By Theorem 5, G is connected. Since G has $2n$ vertices, there would be a vertex u of the n-clique that is adjacent to another vertex of G, and u would have a degree greater than $n-1$, which is a contradiction. Thus, the clique size $n-1$ is the maximum possible. □

3.1. Upper Bound

The following theorem provides a polynomially computable upper bound on the clique number of graphs from $G_{m,n}$.

Theorem 10.

Let $G=(V,E)\in G_{m,n}$, $n>m\ge 2$. Then

$$\omega \left(G\right)\le \beta \left(G\right)=max\left\{{\beta }_1\left(r\right)=⌊\frac{1+\sqrt{4n(n-1)-8r+1}}{2}⌋,{\beta }_2\left(r\right)=⌊2\sqrt{r}⌋\right\}.$$

(1)

The bound is computable in polynomial time and space and is sharp, e.g., for $r=\frac{n(n-1)}{2}$ for all $G\in G_{n,n}$; for $r=\lfloor n/2\rfloor$ ($n\ge 3$, odd) for all $G\in G_{m,n}$; and for $r=2$ and $G\in G_{2,3}$.

Proof. 

Denote $\mathcal{R}=\left\{\right(u,v)\in G:d(u)=m-1,d(v)=n-1\}$. We have $\left|\mathcal{R}\right|=r$. In the following, we are concerned with the edges that can form a maximum clique Q. We distinguish between two possibilities.

Case 1:

All vertices of Q are from $K_m$ or from $K_n$, but not from both.

Case 2:

Q includes vertices from both $K_m$ and $K_n$.

Consider Case 1. Clearly, the edges from $K_m$ and $K_n$, which have been switched, do not exist anymore in $K_m$ and $K_n$ and thus cannot be edges of Q. Then Q would be of the maximum possible size if all other edges of $K_m$ or $K_n$ form a clique. If $n>m$, these must be in $K_n$. If $n=m$, these can be in each of the two copies of $K_n$. Denote $\left|Q\right|=q$. We consecutively obtain

$$\frac{q(q-1)}{2}\le \frac{n(n-1)}{2}-r,\mathrm{i}.\mathrm{e}.,q^2-q-n(n-1)+2r\le 0.$$

Solving this quadratic inequality for q, for the size of the maximum clique Q we obtain

$$\left|Q\right|\le {\beta }_1\left(r\right)=⌊\frac{1+\sqrt{4n(n-1)-8r+1}}{2}⌋.$$

The minimum possible value of ${\beta }_1\left(r\right)$ is achieved for $r=\frac{n(n-1)}{2}$, and equals 1. Note that this is the value of the maximum clique within the vertices of the original clique $K_n$ if all its edges are switched with edges of $K_m$ provided that $m=n$. Denote $K_n^{\prime }=K_n$ and $K_n^{″}=K_m=K_n$. Then both $G\left[K_n^{\prime }\right]$ and $G\left[K_n^{″}\right]$ are empty graphs (i.e., with no edges). Apart from this degenerate case, the graph $G\left[K_n^{\prime }\right]$ is nonempty, with at least one edge, thus having a clique of size 2.

The maximum possible value of ${\beta }_1\left(r\right)$ is achieved for $r=2$ (i.e., if only one 2-switching between edges of $K_m$ and $K_n$ is made). In that case we have ${\beta }_1\left(2\right)=\frac{1+\sqrt{4n(n-1)-15}}{2}$. It is easy to see that for $n\ge 3$, $n-1\le \frac{1+\sqrt{4n(n-1)-15}}{2}<n$, i.e., ${\beta }_1\left(2\right)=n-1$. By Theorem 9, this is the maximum possible clique in a graph of $G_{m,n}$, i.e., ${\beta }_1\left(2\right)=\omega \left(G\right)=n-1$.

Now consider Case 2. It is not hard to see that Q would have a maximum possible size if all vertices of a subclique $K_{p_1}$ of $K_m$ are connected by edges from $\mathcal{R}$ to all vertices of a subclique $K_{p_2}$ of $K_n$. Obviously, a subset of $\mathcal{R}$ of $p_1{p}_2$ edges are needed to realize those connections, as each of these edges is incident to a vertex of Q and a vertex not in Q. Thus, we must have $p_1{p}_2\le r$, as $\left|Q\right|=p_1+p_2$. Let $p=max\{p_1,p_2\}$. Then $p_1{p}_2\le r$ if $p^2\le r$, i.e., if $p\le \sqrt{r}$. Thus, for the size of the maximum clique Q, we obtain

$$\left|Q\right|=p_1+p_2\le {\beta }_2\left(r\right)=\lfloor 2\sqrt{r}\rfloor .$$

The minimum possible value of ${\beta }_2\left(r\right)$ is obviously reached for $r=2$ and we have ${\beta }_2\left(2\right)=\lfloor 2\sqrt{2}\rfloor =2$. This value is achieved, for example, if G is degree-equivalent to $K_3\cup K_2$. In that case, ${\beta }_2\left(2\right)=\omega \left(G\right)=2$.

The maximum possible value of ${\beta }_2\left(r\right)$ is achieved for $r=p^2=p_1{p}_2$, where $p=p_1=p_2=\lfloor n/2\rfloor$ and n is an odd number greater than or equal to 3. Then ${\beta }_2\left(r\right)=\lfloor 2\sqrt{{\left(\lfloor n/2\rfloor \right)}^2}\rfloor =\lfloor 2\lfloor \frac{n}{2}\rfloor \rfloor =2\lfloor \frac{n}{2}\rfloor =n-1$.

Obviously, the bound (1) is efficiently computable. □

Corollary 1.

Under the conditions of Theorem 10, $\alpha \left(\overline{G}\right)\le \beta \left(r\right)$.

Following from the well-known fact that Q is a clique in a graph G if and only if it is an independent set in its complement $\overline{G}$.

3.2. Lower Bound

The following theorem provides a polynomially computable lower bound on the clique number of graphs from $G_{m,n}$.

Theorem 11.

Let $G=(V,E)\in G_{m,n}$, $n>m\ge 2$. Then

$$\omega \left(G\right)\ge \zeta \left(r\right)=⌊\frac{\sqrt{4n(n-1)/r+1}+1}{2}⌋.$$

(2)

The bound is computable in polynomial time and space and is sharp, e.g., for $r=\frac{n(n-1)}{2}$ for all $G\in G_{n,n}$, and for $r=2$ and $G\in G_{3,4}$.

Proof. 

An edge $(u,v)$ with $deg\left(u\right)=m-1$, $deg\left(v\right)=n-1$ is not in the original clique $K_n$, and it corresponds to an edge that has been removed from $K_n$ as a result of a 2-switching. Let e be an edge of $K_n$ that has been switched with an edge of $K_m$. It is not hard to realize that any edge of $K_n$ belongs to $\left(\genfrac{}{}{0pt}{}{n-2}{k-2}\right)$ subcliques of size k within $K_n$. Hence, when e is removed from $K_n$, $\left(\genfrac{}{}{0pt}{}{n-2}{k-2}\right)$ subcliques of size k are removed from $K_n$, as well. Note also that different edges can belong to the same subclique of $K_n$. Hence, the removal of r edges from $K_n$ causes the removal of no more than $r\left(\genfrac{}{}{0pt}{}{n-2}{k-2}\right)$ subcliques of size k from $K_n$. The total number of k-cliques ($k<n$) in $K_n$ is $\left(\genfrac{}{}{0pt}{}{n}{k}\right)$. Hence, if

$$r\left(\genfrac{}{}{0pt}{}{n-2}{k-2}\right)<\left(\genfrac{}{}{0pt}{}{n}{k}\right),$$

(3)

at least one k-clique would be present in $K_n$.

Thus, we obtain that $\omega \left(G\right)\ge max\left\{k:r\left(\genfrac{}{}{0pt}{}{n-2}{k-2}\right)<\left(\genfrac{}{}{0pt}{}{n}{k}\right)\right\}$. Solving inequality (3) for k, we obtain that the maximum value of k satisfying (3) equals

$$\zeta \left(r\right)=⌊\frac{\sqrt{4n(n-1)/r+1}+1}{2}⌋.$$

Note that r cannot exceed $\frac{m(m-1)}{2}\le \frac{n(n-1)}{2}$, and $r=\frac{n(n-1)}{2}$ is possible only if $m=n$. If $r=\frac{n(n-1)}{2}$, then $r\left(\genfrac{}{}{0pt}{}{n-2}{k-2}\right)=\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ if and only if $k=2$. It is easy to check that if $r<\frac{n(n-1)}{2}$ and $k=2$, then $r\left(\genfrac{}{}{0pt}{}{n-2}{k-2}\right)<\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ always holds. This is in accordance with the fact that a 2-clique in G always exists.

Clearly, the bound (2) is computable in polynomial time and space. The example in Figure 4 shows that it is sharp for $G\in G_{3,4}$. □

Corollary 2.

Under the conditions of Theorem 11, $\alpha \left(\overline{G}\right)\ge \zeta \left(r\right)$.

This follows by the same argument as Corollary 1.

Combining the outcomes of Corollaries 1 and 2, we obtain the following result about the vertex cover number of graphs in ${\overline{G}}_{m,n}$.

Corollary 3.

Under the conditions of Theorems 10 and 11,

$$\left|V\right|-\zeta \left(r\right)\ge \tau \left(\overline{G}\right)\ge \left|V\right|-\beta \left(r\right).$$

Proof. 

It is well known that for any graph $G=(V,E)$, $\alpha \left(G\right)+\tau \left(G\right)=\left|V\right|$. Thus, for a graph $G\in G_{m,n}$, we have $\alpha \left(\overline{G}\right)+\tau \left(\overline{G}\right)=\left|V\right|$. This identity, coupled with Corollary 1, implies $\tau \left(\overline{G}\right)=\left|V\right|-\alpha \left(\overline{G}\right)\ge \left|V\right|-\beta \left(r\right)$. The same identity, coupled with Corollary 2, implies $\tau \left(\overline{G}\right)\le \left|V\right|-\zeta \left(r\right)$. □

4. Bounds on the Independence Number

Next, we provide upper and lower bounds on the independence number of graphs in $G_{m,n}$.

4.1. Upper Bound

The following theorem provides a polynomially computable upper bound on the independence number of graphs from $G_{m,n}$.

Theorem 12.

Let $G\in G_{m,n}$ and let G have p vertices of degree $m-1$ connected by edges to q vertices of degree $n-1$. Then $\alpha \left(G\right)\le \eta (p,q)=min\{p,q\}+1\le r+1$. The bound is computable in polynomial time and space and is sharp, e.g., for $\eta (p,q)=r=2$ and $G\in G_{3,4}$.

Proof. 

We consider separately two cases:

Case 1:

There is a maximum independent set of G which has vertices of the same degree (Figure 5, left).

Case 2:

Every maximum independent set of G has vertices of degree $m-1$ and of degree $n-1$ (Figure 5, middle, right).

Let’s consider Case 1. Let S be such a maximum independent set with $\left|S\right|=s\le p$. W.l.o.g., suppose that its vertices are in $K_m$. Each of these vertices is an endpoint of an edge whose other endpoint is in $K_n$. Set S is independent, i.e., with no edges among its elements (so, it is a clique in $\overline{G}$). The set of edges $E_1$ connecting the vertices of S has been removed from $K_m$ by a series of 2-switchings with a set $E_2$ of the same number of edges from $K_n$ (the latter is also being removed from $K_n$). Before the removal, the edges in $E_2$ have been incident to a set T of at least s vertices of degree $n-1$ (we would have $\left|T\right|=s$, if $E_2$ is an s-clique within $K_n$ and which, after the removal of $E_2$, becomes an independent set of s vertices of degree $n-1$). Clearly, if $E_2$ is not a clique, its edges would be incident to more than s vertices of degree $n-1$, and after the removal of $E_2$, the vertices in T will be adjacent to vertices of degree $m-1$. Hence, $q\ge \left|T\right|\ge s$.

Thus, we have $\alpha \left(G\right)\le p$ and $\alpha \left(G\right)\le q$, i.e., $\alpha \left(G\right)\le min\{p,q\}\le r$, as $min\{p,q\}=r$ only if the edges whose vertices have different degrees form a matching (i.e., no two of these edges share a vertex).

Now let’s consider Case 2. Let S be such a maximum independent set, as $S=S_1\cup S_2$, $S_1\subseteq K_m$, $S_2\subseteq K_n$. If G is different from $K_m\cup K_n$, Caro-Wei’s theorem (Theorem 3) implies $\alpha \left(G\right)\ge 3$. Then, at least one of the sets $S_1$ and $S_2$ has at least two elements. Let $|S_1|\ge 2$. Set $S_1$ is independent, which means that all edges between its elements have been removed from $K_m$ by 2-switchings with some edges of $K_n$. Thus, each of these vertices is connected by at least one edge to a vertex from $K_n$. Regarding the vertices of $S_2$, we distinguish between two possibilities:

(a) The vertices of $S_1$ are connected to vertices of $K_n$ and the vertices of $S_2$ are connected to vertices of $K_m$ (Figure 5, middle).

Let $A_1$ be the set of vertices to which the vertices of $S_1$ are connected, and $A_2$ be the set of vertices to which the vertices of $S_2$ are connected. By the same argument as in Case 1, we have that $|S_1| \le |A_1|$ and $|S_2| \le |A_2|$. Now if $|S_1| + |A_2| \le |S_2| + |A_1|$, then $|S_1| + |S_2| \le |S_1| + |A_2|$, and if $|S_2| + |A_1| \le |S_1| + |A_2|$, then $|S_1| + |S_2| \le |S_2| + |A_1|$, i.e., in either case, the size of the independent set does not exceed the minimum number of vertices of the same degree, which are connected to vertices of another degree.

(b) The vertices of $S_1$ are connected to vertices of $K_n$ while the vertices of $S_2$ are not connected to vertices of $K_m$, or vice-versa (Figure 5, right).

It is easy to see that this is possible only if $S_2$ contains a single vertex v. (Otherwise, all edges in $K_n$ between the vertices of $S_2$ are removed in G; this is possible only if 2-switchings with edges from $K_m$ have been made, which would cause new edges linking vertices from $S_2$ to vertices of $K_m$.) Thus, for the maximum independent set, we have $S=S_1\cup \left\{v\right\}$.

For the number of vertices in $S_1$ we have the bound from Case 1, i.e., $|S_1| =min\{p,q\}$. Then $\left|S\right|=\eta (p,q)=min\{p,q\}+1$.

It is easy to see that the bound is computable in $O\left(\right|E\left|\right)$ time and space. Moreover, for the graph in Figure 5, right, the bound of the theorem is reached for $G\in G_{3,4}$. □

Corollary 4.

Under the conditions of Theorem 12, $\omega \left(\overline{G}\right)\le \eta (p,q)\le r+1$.

4.2. Lower Bounds

Numerous upper and lower bounds on the independence number are available in the literature. The most important of these are listed in the brief survey [25]. As a rule, the lower bounds for general graphs applied to the specific classes of graphs considered in the present paper guarantee that the independence number is at least 3. The following proposition shows that under certain conditions, $\alpha \left(G\right)$ and $\alpha \left(\overline{G}\right)$ can be at least $\frac{logn}{2log2}$.

Proposition 1.

Under the conditions of Theorems 10 and 12,

if $\beta \left(r\right)<\frac{logn}{2log2}$, then $\alpha \left(G\right)\ge \frac{logn}{2log2}$;

if $\eta (p,q)<\frac{logn}{2log2}$, then $\alpha \left(\overline{G}\right)\ge \frac{logn}{2log2}$.

Proof. 

Let us first note that, since each clique in a graph G on n vertices is an independent set in its complement $\overline{G}$, Theorem 4 holds if one defines $A\left(n\right)$ to be the greatest integer such that either G or $\overline{G}$ contains an independent set of order $A\left(n\right)$. Thus, for the so-defined independent set of size $A\left(n\right)$ we have that if $A\left(n\right)\ge 3$, then $\frac{logn}{2log2}<A\left(n\right)<\frac{2logn}{log2}$. Note also that an independence set in any graph $G\in G_{m,n}$ has a size of at least 3, which follows from Theorem 3.

From Corollary 1, we have $\alpha \left(\overline{G}\right)\le \beta \left(r\right)$. Then if $\beta \left(r\right)<\frac{logn}{2log2}$, by the independence number version of Theorem 4 it follows that that $\alpha \left(G\right)\ge \frac{logn}{2log2}$. Likewise, from Theorem 12, we have $\alpha \left(G\right)\le \eta (p,q)$. Then, if $\eta (p,q)<\frac{logn}{2log2}$, by the independence number version of Theorem 4 we have $\alpha \left(\overline{G}\right)\ge \frac{logn}{2log2}$. □

Along with the well-known upper and lower bounds for the independence number of a graph, the survey [25] also lists several conjectures made by the conjecture-making software Graffiti (see [26]) and AGX (see [27]). In the rest of this section, we show that some of these conjectures trivially hold for graphs studied in this paper. As a matter of fact, in all validations, one can use Caro-Wei’s lower bound given in Theorem 3. Recall that this bound implies that a graph G that is degree-equivalent to $K_m\cup K_n$ has independence number $\alpha \left(G\right)\ge 2$, as equality holds only if $G=K_m\cup K_n$. Thus, for a graph $G\in G_{m,n}$, $G\ne K_m\cup K_n$, $\alpha \left(G\right)\ge 3$ (for a direct proof that $G\in G_{a_1,a_2,\cdots ,a_k}$, $G\ne {\cup }_{i=1}^k{K}_{a_i}$ implies $\alpha \left(G_{a_1,a_2,\cdots ,a_k}\right)\ge k+1$ see [28]). Below, we verify that all conjectured lower bounds do not exceed 3 for graphs from $G_{m,n}$ and their complements.

Conjecture 1.

$\alpha \left(G\right)\ge \overline{d}$, where $\overline{d}$ is the average distance between distinct vertices of G (conjectured by Graffiti).

Follows for $G_{n,n}$ and ${\overline{G}}_{n,n}$ from the bounds on these graphs diameter given in Theorems 7 and 8, and the inequalities $\overline{d}\le 3\le \alpha \left(G\right)$ which hold for these graphs.

Conjecture 2.

$\alpha \left(G\right)\ge r\left(G\right)$, where $r\left(G\right)$ is the graph radius (conjectured by Graffiti).

Follows for $G_{n,n}$ and ${\overline{G}}_{n,n}$ from the bounds of Theorems 7 and 8 and the inequalities $r\left(G\right)\le diam\left(G\right)\le 3\le \alpha \left(G\right)$ which hold for these graphs.

Conjecture 3.

$\alpha \left(G\right)\ge 1+c\left(G\right)/2$, where $c\left(G\right)$ is the number of cut-vertices of G (conjectured by Graffiti).

Follows for $G_{m,n},G_{n,n},{\overline{G}}_{n,n}$, and ${\overline{G}}_{m,n}$ from Theorem 6 and the inequalities $1+c\left(G\right)/2<3\le \alpha \left(G\right)$ which hold for these graphs.

Conjecture 4.

$\alpha \left(G\right)\ge \lceil 2\sqrt{v}\rceil -\Delta$, where v is the number of the graph vertices and Δ is the maximum degree of the graph vertices (conjectured by the AGX conjecture-making software).

Follows straightforwardly for $G_{m,n},G_{n,n},{\overline{G}}_{n,n}$, and ${\overline{G}}_{m,n}$ from the inequalities $\lceil 2\sqrt{v}\rceil -\Delta \le 3\le \alpha \left(G\right)$, where G is in one of the above four classes. For example, setting $v=m+n$, for $G\in {\overline{G}}_{m,n}$ we have: $\lceil 2\sqrt{v}\rceil -\Delta =\lceil 2\sqrt{m+n}\rceil -n\le 2\sqrt{2n}+1-n\le 3\le \alpha \left(G\right)$ for all positive integers $m,n$. For $G\in G_{m,n}$ we have: $\lceil 2\sqrt{v}\rceil -\Delta =\lceil 2\sqrt{m+n}\rceil -n+1\le 2\sqrt{2n}+2-n\le 3\le \alpha \left(G\right)$ for all integers $n\ge 6$.

Conjecture 5.

$\alpha \left(G\right)\ge \frac{v-1}{\Delta }$, where v and Δ are as in the previous conjecture (conjectured by the AGX conjecture-making software).

Holds for $G\in G_{m,n},G_{n,n},{\overline{G}}_{n,n}$, and ${\overline{G}}_{m,n}$. For example, if $G\in G_{m,n}$, setting $v=m+n$, we have: $\frac{v-1}{\Delta }=\frac{m+n-1}{n-1}\le \frac{2n-1}{n-1}\le 3\le \alpha \left(G\right)$ for all positive integers $m,n$.

5. Regular Graphs

In this section, we consider the regular graphs $G_{d,\cdots ,d}$ which are degree-equivalent to k disjoint cliques $K_d$, and which, unlike the graphs from $G_{m,n}$, can be sparse, e.g., for a large k and a bounded d. It is easy to see that a d-regular graph $G=(V,E)$ is degree-equivalent to a graph $U_{i=1}^k{K}_{d+1}$ if and only if for some positive integer k the following condition holds:

$$\left|V\right|=(d+1)k$$

(4)

For the clique number of such graphs, we have the trivial sharp bounds $2\le \omega \left(G\right)\le d$, while for the independence number, it trivially follows that $\alpha \left(G\right)\le ⌊\frac{n}{2}⌋$.

Several well-known lower bounds for general graphs [29,30,31] when applied to $G_{d,\cdots ,d}$ imply $\alpha \left(G\right)\ge k$, while Caro-Wei’s theorem (Theorem 3) implies that $\alpha \left(G\right)\ge k+1$ for any graph G that is degree-equivalent to k cliques. Below, we give a simple direct proof for the case of a regular graph.

Proposition 2.

If $G\ne {\bigcup }_{i=1}^k{K}_n$ and $D\left(G\right)=D\left({\bigcup }_{i=1}^k{K}_n\right)$, then $\alpha \left(G\right)\ge k+1$, as the bound is sharp.

Proof. 

Since $D\left(G\right)=D\left({\bigcup }_{i=1}^k{K}_n\right)$, the degree of every vertex in G must be $n-1$ and G must have $kn$ vertices. Suppose $\alpha \left(G\right)\le k$ and let $u_1,\dots ,u_j$ be a maximum independent set in G, $j\le k$. We have

$$|\bigcup _{i=1}^{j}N\left(u_i\right)|\le \sum _{i=1}^j\left|N\left(u_i\right)\right|=j(n-1)=jn-j\le kn-j,$$

(5)

where the first inequality is a basic fact in set theory, the first equality follows because $\left|N\right(u_i\left)\right|=n-1$ for each $1\le i\le j$, and the second inequality follows because $j\le k$.

If $|{\bigcup }_{i=1}^{j}N\left(u_i\right)| <kn-j$, then there must be a vertex v which is not adjacent to any of $u_1,\dots ,u_j$, so $\{v,u_1,\dots ,u_j\}$ is an independent set, contradicting that $\{u_1,\dots ,u_j\}$ is a maximum independent set.

If $|{\bigcup }_{i=1}^{j}N\left(u_i\right)| =kn-j$, then both inequalities in (5) must be equalities, so $j=k$ and $|{\bigcup }_{i=1}^{j}N\left(u_i\right)| ={\sum }_{i=1}^j\left|N\left(u_i\right)\right|$, which implies that $N\left(u_1\right)\dots N\left(u_j\right)$ are pairwise disjoint. Now, if $G\left[N\left(u_{\ell }\right)\right]$ is not a clique for some $1\le \ell \le j$, then there are two vertices v and w in $N\left(u_{\ell }\right)$ which are not adjacent. Then, $\{u_1,\dots ,u_{\ell -1},v,w,u_{\ell +1},\dots ,u_j\}$ is an independent set of size $j+1$, contradicting that $\{u_1,\dots ,u_j\}$ is a maximum independent set. Thus, $G\left[N\left(u_i\right)\right]$ must be a clique for each $1\le i\le j$ (and hence also $G[N\left(u_i\right)\cup u_i]$ must be a clique for each $1\le i\le j$). But since $j=k$, this means there are k cliques of size n in G, which contradicts the fact that $G\ne {\bigcup }_{i=1}^k{K}_n$ and completes the proof.

This bound is sharp (for example, a 6-cycle, which is degree-equivalent to $K_3\cup K_3$ has independence number 3). □

$G_{d,\cdots ,d}$ Graphs of Bounded Degree

Graph theory is regarded as a key tool in modeling and solving practical problems involving big data. In his recent monograph devoted to the matter [21], Lovász emphasizes the special importance of investigating properties of graphs of bounded degree, as most large networks arising in real-life applications are of that kind. In this section, we consider such sort of graphs $G_{d,\cdots ,d}$.

Consider a graph $G\in G_{d,\cdots ,d}\simeq {\bigcup }_{i=1}^k{K}_d$ on $n=kd$ vertices, where d is a constant, i.e., $k=\mathsf{\Omega }\left(n\right)$. By Caro-Wei’s Theorem (Theorem 3) we have $\alpha \left(G\right)\ge k+1=\mathsf{\Omega }\left(n\right)$. Thus, we have found that under the conditions considered, the graph G is very stable. In particular, for cubic graphs (i.e., regular graphs whose all vertices are of degree three), the well-known Brooks theorem [32] implies that $\alpha \left(G\right)\ge n/3$. (Brooks theorem states that for any connected undirected graph G with maximum degree $\Delta$, the chromatic number of G is at most $\Delta$, unless G is a complete graph or an odd cycle, in which case the chromatic number is $\Delta +1$.) In the rest of this section, we present some other properties of the class of cubic graphs $G_{4,\cdots ,4}\simeq {\bigcup }_{i=1}^k{K}_3$, as well as of the 4-regular graphs $G_{5,\cdots ,5}$.

Proposition 3.

All connected cubic graphs in $G_{4,\cdots ,4}$ are planar.

Proof. 

Let $G\in G_{4,\cdots ,4}$. Clearly, G does not contain $K_5$ as a subgraph. Also, if G is connected, it cannot have $K_{3,3}$ as a proper subgraph since otherwise it will have at least one vertex of degree greater than 3. Moreover, $K_{3,3}$ itself is not in $G_{4,\cdots ,4}$ since obviously its degree sequence does not match the one of any union of $K_4$’s, so $G\ne K_{3,3}$, which completes the proof. □

Remark 1.

A disconnected graph $G\in G_{4,\cdots ,4}$ can be non-planar, as it can contain $K_{3,3}$ as a component. For example, the graph $G=K_{3,3}\cup K_{3,3}$ has the same degree sequence as $H=K_4\cup K_4\cup K_4$, and by Hakimi’s theorem, G can be obtained from H by a finite sequence of 2-switchings.

In complexity theory, cubic and 4-regular graphs are the best-studied regular graphs. It is well known that problems such as finding a maximum independence set or minimum vertex cover (as well as computing the maximum independence number and minimum vertex cover number) are NP-complete on planar cubic graphs, while the problem of finding a minimum domination set or minimum graph coloring (as well as computing the domination number and the chromatic number) are NP-complete on 4-regular planar graphs [16,33]. The following proposition implies that the same holds for cubic $G_{4,\cdots ,4}$ graphs and 4-regular $G_{5,\cdots ,5}$ graphs.

Proposition 4.

The problem of finding the independence number $\alpha \left(G\right)$ and the vertex cover number $\tau \left(G\right)$ are NP-complete on cubic $G_{4,\cdots ,4}$ graphs. The problem of finding the domination number $\gamma \left(G\right)$ and the chromatic number $\chi \left(G\right)$ are NP-complete on 4-regular $G_{5,\cdots ,5}$ graphs.

Proof. 

It is trivial to see that on the specified classes of graphs, each of these problems is in $NP$: given a positive integer B and a candidate solution I, one can check in polynomial time if I is, say, an independent set of G and if $\left|I\right|\le B$.

Let $G^{\prime }=(V^{\prime },E^{\prime })$ be a planar cubic graph and denote $|V^{\prime }| =n^{\prime }$. Consider the graph $G=(V,E)$ with 4 components that are copies of $G^{\prime }$. G has $4n^{\prime }$ vertices, and it has the same degree sequence as a graph H with $n^{\prime }$ components $K_4$. By Hakimi’s theorem (Theorem 1), it can be obtained from H by a finite number of 2-switchings and thus $G\in G_{4,\cdots ,4}\simeq {\bigcup }_{i=1}^{n^{\prime }}{K}_4$. We clearly have $\alpha \left(G\right)=4\alpha \left(G^{\prime }\right)$ and $\tau \left(G\right)=4\tau \left(G^{\prime }\right)$. Since the computations of the independence number and the vertex cover number are NP-hard on planar cubic graphs [16], the above equalities imply that the same holds for cubic $G_{4,\cdots ,4}$ graphs.

Now let $G^{\prime }=(V^{\prime },E^{\prime })$ be a 4-regular planar graph with $|V^{\prime }| =n^{\prime }$. The same construction as above, using $K_5$ instead of $K_4$, yields a graph G that is degree-equivalent to a graph H with $n^{\prime }$ components $K_5$. Hence, $G\in G_{5,\cdots ,5}\simeq {\bigcup }_{i=1}^{n^{\prime }}{K}_5$. Obviously, $\gamma \left(G\right)=4\gamma \left(G^{\prime }\right)$ and $\chi \left(G\right)=\chi \left(G^{\prime }\right)$. Since the computations of the domination number and the chromatic number are NP-hard on planar 4-regular graphs [16], the same holds for 4-regular $G_{5,\cdots ,5}$ graphs. □

6. Conclusions

In this paper, we obtained upper and lower bounds for the clique number and the independence number of graphs, which are degree-equivalent to two disjoint cliques. We also studied some properties of regular (in particular, cubic and 4-regular) graphs that are degree-equivalent to arbitrary by many cliques. Further research will aim at extending some of these results to arbitrary graphs that are degree-equivalent to more than two cliques. Regarding practical applications, it would be interesting to study more in detail subclasses of sparse graphs $G_{p_1,\cdots ,p_k}$, in particular, cubic and 4-regular graphs. Examining multigraphs of that type is seen as another challenging task.