The Hadwiger number, chordal graphs and $ab$-perfection

TL;DR

This paper characterizes chordal graphs and various graph families based on equalities involving the Hadwiger number and other graph invariants, providing new insights into graph structure and properties.

Contribution

It introduces characterizations of chordal graphs and specific graph families through equalities involving the Hadwiger number and other invariants, expanding understanding of graph perfection.

Findings

Chordal graphs characterized by Hadwiger number properties.

Families of graphs where Hadwiger number equals clique or chromatic numbers.

New relationships between Hadwiger number and various graph invariants.

Abstract

A graph is chordal if every induced cycle has three vertices. The Hadwiger number is the order of the largest complete minor of a graph. We characterize the chordal graphs in terms of the Hadwiger number and we also characterize the families of graphs such that for each induced subgraph $H$, (1) the Hadwiger number of $H$ is equal to the maximum clique order of $H$, (2) the Hadwiger number of $H$ is equal to the achromatic number of $H$, (3) the $b$-chromatic number is equal to the pseudoachromatic number, (4) the pseudo-$b$-chromatic number is equal to the pseudoachromatic number, (5) the Hadwiger number of $H$ is equal to the Grundy number of $H$, and (6) the $b$-chromatic number is equal to the pseudo-Grundy number.

Figures

Equations

$$\omega (G)\le h(G)$$

$$\omega (G)=\max \{\omega (H_1),\omega (H_2)\}=k.$$

$$\max \{\omega (H_1),\omega (H_2)\}=\max \{h(H_1),h(H_2)\}=k.$$

$$h(G)\le \max \{h(H_1),h(H_2)\}=k.$$

$$\omega (G)\le h(G)\le \psi (G).$$

$$\omega (G)\le \alpha (G)\le \psi (G).$$

$$\omega (G)\le b(G)\le B(G)\le \psi (G).$$

$$\omega (G)\le \Gamma (G)\le \gamma (G).$$

Full text

Keywords: Complete colorings, perfect graphs, forbidden graphs characterization.

2010 Mathematics Subject Classification: 05C17; 05C15; 05C83.

1 Introduction

Let $G$ be a finite graph. A $k$-coloring of $G$ is a surjective function $\varsigma$ that assigns a number from the set $[k]:=\{1,\dots,k\}$ to each vertex of $G$. A $k$-coloring $\varsigma$ of $G$ is called proper if any two adjacent vertices have different colors, and $\varsigma$ is called complete if for each pair of different colors $i,j\in[k]$ there exists an edge $xy\in E(G)$ such that $x\in\varsigma^{-1}(i)$ and $y\in\varsigma^{-1}(j)$. A $k$-coloring $\varsigma$ of a connected graph $G$ is called connected if for all $i\in[k]$, each color class $\varsigma^{-1}(i)$ induces a connected subgraph of $G$.

The chromatic number $\chi(G)$ of $G$ is the smallest number $k$ for which there exists a proper $k$-coloring of $G$. The Hadwiger number $h(G)$ is the maximum $k$ for which a connected and complete coloring of a connected graph $G$ exists, and it is defined as the maximum $h(H)$ among the connected components $H$ of a disconnected graph $G$ (it is also known as the connected-pseudoachromatic number, see [1]).

A graph $H$ is called a minor of the graph $G$ if and only if $H$ can be formed from $G$ by deleting edges and vertices and by contracting edges. Suppose that $K_{k}$ is a minor of a connected graph $G$. If $V(K_{k})=[k]$ then there exists a natural corresponding complete $k$-coloring $\varsigma\colon G\rightarrow[k]$ for which $\varsigma^{-1}(i)$ is exactly the set of vertices of $G$ which contract to vertex $i$ in $K_{k}$. The Hadwiger number $h(G)$ of a graph $G$ is the largest $k$ for which $K_{k}$ is a minor of $G$. Clearly,

[TABLE]

where $\omega(G)$ denotes the clique number of $G$: the maximum clique order of $G$.

The Hadwiger number was introduced by Hadwiger in 1943 [14] together the Hadwiger conjecture which states that $\chi(G)\leq h(G)$ for any graph $G$.

The following definition is an extension of the notion of perfect graph, introduced by Berge [4]: Let $a,b$ be two distinct parameters of $G$. A graph $G$ is called $ab$-perfect if for every induced subgraph $H$ of $G$, $a(H)=b(H)$. Note that, with this definition a perfect graph is denoted by $\omega\chi$-perfect. The concept of the $ab$-perfect graphs was introduced by Christen and Selkow in [8] and extended in [3, 2, 6, 18, 19, 20].

A graph $G$ without an induced subgraph $H$ is called $H$-free. A graph $H_{1}$-free, $H_{2}$-free,… is called $(H_{1},H_{2},\dots)$-free. A chordal graph is a $(C_{4},C_{5},\dots)$-free one.

Some known results are the following: Lóvasz proved in [17] that a graph $G$ is $\omega\chi$-perfect if and only if its complement is $\omega\chi$-perfect. Chudnovsky, Robertson, Seymour and Thomas proved in [9] that a graph $G$ is $\omega\chi$-perfect if and only if $G$ and its complement are $(C_{5},C_{7},\dots)$-free.

This paper is organized as follows: In Section 2 we prove that the families of chordal graphs and the family of $\omega h$-perfect graphs are the same. In Section 3, we give some consequences of the Section 2 as characterizations of other graph families related to complete colorings.

2 Chordal graphs and $\omega h$-perfect graphs

We will use the following chordal graph characterization to prove Theorem 2.2:

Theorem 2.1** (Hajnal, Surányi [15] and Dirac [10]).**

A graph $G$ is chordal if and only if $G$ can be obtained by identifying two complete subgraphs of the same order in two chordal graphs.

Now, we characterize the chordal graphs and the $\omega h$-perfect ones. The following proof is based on the standard proof of the chordal graph perfection (see [7]).

Theorem 2.2**.**

A graph $G$ is $\omega h$-perfect if and only if $G$ is chordal.

Proof.

Assume that $G$ is $\omega h$-perfect. Note that if a cycle $H$ is one of four or more vertices then $\omega(H)=2$ and $h(H)=3$. Hence, every induced cycle of $G$ has at the most $3$ vertices and the implication is true.

Now, we verify the converse. Since every induced subgraph of a chordal graph is also a chordal graph, it suffices to show that if $G$ is a connected chordal graph, then $\omega(G)=h(G)$. We proceed by induction on the order $n$ of $G$. If $n=1$, then $G=K_{1}$ and $\omega(G)=h(G)=1$. Assume, therefore, that $\omega(H)=h(H)$ for every induced chordal graph $H$ of order less than $n$ for $n\geq 2$ and let $G$ be a chordal graph of order $n$. If $G$ is a complete graph, then $\omega(G)=h(G)=n$. Hence, we may assume that $G$ is not complete. By Theorem 2.1, $G$ can be obtained from two chordal graphs $H_{1}$ and $H_{2}$ by identifying two complete subgraphs of the same order in $H_{1}$ and $H_{2}$. Let $S$ denote the set of vertices in $G$ that belong to $H_{1}$ and $H_{2}$. Thus the induced subgraph $\left\langle S\right\rangle_{G}$ in $G$ by $S$ is complete and no vertex in $V(H_{1})\setminus S$ is adjacent to a vertex in $V(H_{2})\setminus S$. Hence,

[TABLE]

Moreover, according to the induction hypothesis, $\omega(H_{1})=h(H_{1})$ and $\omega(H_{2})=h(H_{2})$, then

[TABLE]

On the other hand, since $S$ is a clique cut then each walk between $V(H_{1})\setminus S$ and $V(H_{2})\setminus S$ contains at least one vertex in $S$. Let $\varsigma$ be a pseudo-connected $h(G)$-coloring of $G$, and suppose there exist two color classes such that one is completely contained in $V(H_{1})\setminus S$, and the other one is completely contained in $V(H_{2})\setminus S$. Clearly these two color classes do not intersect, which contradicts our choice of $\varsigma$. Moreover, each color class with vertices both in $V(H_{1})\setminus S$ and in $V(H_{2})\setminus S$, contains vertices in $S$. Consequently, every pair of color classes having vertices both in $V(H_{1})\setminus S$ and in $V(H_{2})\setminus S$ must have an incidence in $\left\langle S\right\rangle_{G}$. Thus,

[TABLE]

By Equation 1, $\omega(G)=k=h(G)$ and the result follows. ∎

It is known that every chordal graph is a $\omega\chi$-perfect one (see [7]). The following corollary is a consequence of the chordal graph perfection.

Corollary 2.3**.**

Every $\omega h$-perfect graph is $\omega\chi$-perfect.

3 Other classes of $ab$-perfect graphs

In this section, we give a new characterization of several family of $ab$-perfect graphs related to complete colorings.

3.1 Achromatic and pseudoachromatic numbers

Firstly, the pseudoachromatic number $\psi(G)$ of $G$ is the largest number $k$ for which there exists a complete $k$-coloring of $G$ [13], and it is easy to see that

[TABLE]

Secondly, the achromatic number $\alpha(G)$ of $G$ is the largest number $k$ for which there exists a proper and complete $k$-coloring of $G$ [16], and it is not hard to see that

[TABLE]

Complete bipartite graphs have achromatic number two (see [7]) but their Hadwiger number can be arbitrarily large, while the graph formed by the union of $K_{2}$ has Hadwiger number two but its achromatic number can be arbitrarily large. Therefore, $\alpha$ and $h$ are two non comparable parameters. We will use the following characterization in the proof of Corollary 3.2.

Theorem 3.1** (Araujo-Pardo, R-M [3, 2]).**

A graph $G$ is $\omega\psi$-perfect if and only if $G$ is $(C_{4},P_{4},P_{3}\cup K_{2},3K_{2})$-free.

Corollary 3.2 is an interesting result because it gives a characterization of two non comparable parameters.

Corollary 3.2**.**

A graph $G$ is $\alpha h$-perfect if and only if $G$ is $\omega\psi$-perfect.

Proof.

Since $h(C_{4})=\alpha(P_{4})=\alpha(P_{3}\cup K_{2})=\alpha(3K_{2})=3$ and $\alpha(C_{4})=h(P_{4})=h(P_{3}\cup K_{2})=h(3K_{2})=2$ (see Figure 1) then a $\alpha h$-perfect graph is $(C_{4},P_{4},P_{3}\cup K_{2},3K_{2})$-free. By Theorem 3.1, $G$ is $\omega\psi$-perfect.

For the converse, if $G$ is $\omega\psi$-perfect, then by Equation 2, $G$ is a $\omega h$-perfect graph, thus, the implication follows. ∎

Corollary 3.3**.**

Every $\omega\psi$-perfect graph is $\omega\chi$-perfect.

Proof.

If a graph $G$ is $\omega\psi$-perfect then Equation (2) implies that $G$ is $\omega h$-perfect, and by Theorem 2.2 $G$ is chordal, therefore $G$ is $\omega\chi$-perfect. ∎

The following corollary is a consequence of the perfection of $\omega\psi$-perfect graphs.

Corollary 3.4**.**

Every $\alpha h$-perfect graph is $\omega\chi$-perfect.

3.2 $b$-chromatic and pseudo-$b$-chromatic numbers

On one hand, a coloring such that every color class contains a vertex that has a neighbor in every other color class is called dominating. The pseudo-$b$-chromatic number $B(G)$ of a graph $G$ is the largest integer $k$ such that $G$ admits a dominating $k$-coloring.

On the other hand, the $b$-chromatic number $b(G)$ of $G$ is the largest number $k$ for which there exists a proper and dominating $k$-coloring of $G$ [6], therefore

[TABLE]

We get the following characterizations:

Corollary 3.5**.**

For any graph $G$ the following are equivalent: $\left\langle 1\right\rangle$ $G$ is $\omega\psi$-perfect, $\left\langle 2\right\rangle$ $G$ is $b\psi$-perfect, $\left\langle 3\right\rangle$ $G$ is $B\psi$-perfect and $\left\langle 4\right\rangle$ $G$ is $(C_{4},P_{4},P_{3}\cup K_{2},3K_{2})$-free.

Proof.

The proofs of $\left\langle 1\right\rangle\Rightarrow\left\langle 2\right\rangle$ and $\left\langle 2\right\rangle\Rightarrow\left\langle 3\right\rangle$ immediately follow from (4). To prove $\left\langle 3\right\rangle\Rightarrow\left\langle 4\right\rangle$ note that, if $H\in\{C_{4},P_{4},P_{3}\cup K_{2},3K_{2}\}$ then $B(H)\not=\psi(H)$, hence the implication is true, see Figure 1. The proof of $\left\langle 4\right\rangle\Rightarrow\left\langle 1\right\rangle$ is a consequence of Theorem 3.1. ∎

The following corollary is a consequence of Corollaries 3.3 and 3.5.

Corollary 3.6**.**

The $b\psi$-perfect graphs and the $B\psi$-perfect ones are $\omega\chi$-perfect.

Corollary 3.5 is related to the following theorem:

Theorem 3.7** (Christen, Selkow [8] and Blidia, Ikhlef, Maffray [6]).**

For any graph $G$ the following are equivalent: $\left\langle 1\right\rangle$ $G$ is $\omega\alpha$-perfect, $\left\langle 2\right\rangle$ $G$ is $b\alpha$-perfect and $\left\langle 3\right\rangle$ $G$ is $(P_{4},P_{3}\cup K_{2},3K_{2})$-free.

3.3 Grundy and pseudo-Grundy numbers

First, a coloring of $G$ is called pseudo-Grundy if each vertex is adjacent to some vertex of each smaller color. The pseudo-Grundy number $\gamma(G)$ is the maximum $k$ for which a pseudo-Grundy $k$-coloring of $G$ exists (see [5, 7]).

Second, a proper pseudo-Grundy coloring of $G$ is called Grundy. The Grundy number $\Gamma(G)$ (also known as the first-fit chromatic number) is the maximum $k$ for which a Grundy $k$-coloring of $G$ exists (see [7, 12]). From the definitions, we have that

[TABLE]

The following characterization of the graphs call trivially perfect graphs, it will be used in the proof of Corollary 3.9.

Theorem 3.8** (R-M [19]).**

A graph $G$ is $\omega\gamma$-perfect if and only if $G$ is $(C_{4},P_{4})$-free.

It is known that a trivially perfect graph is chordal (see [11]). The following corollary also gives a characterization of two non comparable parameters.

Corollary 3.9**.**

A graph $G$ is $\Gamma h$-perfect if and only if $G$ is $\omega\gamma$-perfect.

Proof.

A $\Gamma h$-perfect graph is $(C_{4},P_{4})$-free because $\Gamma(C_{4})=h(P_{4})=2$ and $\Gamma(P_{4})=h(C_{4})=3$ (see Figure 1) then by Theorem 3.8, $G$ is $\omega\gamma$-perfect.

For the converse, let $G$ be a $\omega\gamma$-perfect graph. If $H$ is an induced graph of $G$, by Equation 5, $\omega(H)=\Gamma(H)$. Since $G$ is a chordal graph, $\omega(H)=h(H)$, so the implication follows. ∎

The following corollary is a consequence of the perfection of $\omega\gamma$-perfect graphs.

Corollary 3.10**.**

Every $\Gamma h$-perfect graph is $\omega\chi$-perfect.

3.4 The $b\gamma$-perfect graphs

Finally, we will use the following characterization of the proof of Theorem 3.12.

Theorem 3.11** (Blidia, Ikhlef, Maffray [6]).**

A graph $G$ is $b\Gamma$-perfect if and only if $G$ is $(P_{4},3P_{3},2D)$-free.

We get the following characterization.

Theorem 3.12**.**

A graph $G$ is $b\gamma$-perfect if and only if $G$ is $(C_{4},P_{4},3P_{3},2D)$-free.

Proof.

Note that, if $H\in\{C_{4},P_{4},3P_{3},2D\}$ then $b(H)\not=\gamma(H)$, hence, the implication is true (see Figure 1).

For the converse, a $(C_{4},P_{4},3P_{3},2D)$-free graph $G$ is $\omega\gamma$-perfect (by Theorem 3.8) and $b\Gamma$-perfect (by Theorem 3.11). Then, for every induced subgraph $H$ of $G$, $\omega(H)=\gamma(H)=\Gamma(H)$ by Equation 5 and $b(H)=\Gamma(H)$. Therefore, $b(H)=\gamma(H)$ and the result follows. ∎