New bounds for the b-chromatic number of vertex deleted graphs

TL;DR

This paper establishes new lower bounds for the b-chromatic number of vertex-deleted subgraphs, focusing on quasi-line, chordal graphs, and graphs with large girth, advancing understanding of graph coloring properties.

Contribution

It provides novel lower bounds for the b-chromatic number after vertex deletion in specific graph classes, extending previous theoretical results.

Findings

Lower bounds for quasi-line graphs

Lower bounds for chordal graphs

Bounds for graphs with large girth

Abstract

A b-coloring of a graph is a proper coloring of its vertices such that each color class contains a vertex adjacent to at least one vertex of every other color class. The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring with k colors. In this work we present lower bounds for the b-chromatic number of a vertex-deleted subgraph of a graph, particularly regarding two important classes, quasi-line and chordal graphs. We also get bounds for the b-chromatic number of G -{x}, when G is a graph with large girth.

Equations

$$b(G-\{x\})\le b(G)+2$$

$$b(G-\{x\})\le b(G)+1$$

$$b(G-\{x\})\le b(G)+1+\sqrt{d(x)-1}$$

$$b(G-\{x\})\le b(G)+1+\sqrt{\omega -1}$$

Full text

New bounds for the b-chromatic number of vertex deleted graphs

Renata DEL-VECCHIO and Mekkia KOUIDER

Abstract. A b-coloring of a graph is a proper coloring of its vertices such that each color class contains a vertex adjacent to at least one vertex of every other color class. The b-chromatic number of a graph is the largest integer $\displaystyle k$ such that the graph has a b-coloring with $\displaystyle k$ colors. In this work we present lower bounds for the b-chromatic number of a vertex-deleted subgraph of a graph, particularly regarding two important classes, quasi-line and chordal graphs.We also get bounds for the b-chromatic number of $\displaystyle G-\{x\}$, when $\displaystyle G$ is a graph with large girth.

Key words: b-coloring, quasi-line graph, chordal graph, girth

Mathematics Subject Classification: 05C15.

1 Introduction

All graphs considered in this work are simple and undirected. Let $\displaystyle G=(V(G),E(G))$ be an undirected graph where $\displaystyle V(G)$ and $\displaystyle E(G)$ are the sets of its vertices and edges, respectively. If $\displaystyle A\subset V(G)$, we denote by $\displaystyle\langle A\rangle$ the induced subgraph generated by $\displaystyle A$. For any vertex $\displaystyle x$ of a graph G, the neighborhood of $\displaystyle x$ is the set $\displaystyle N(x)=\{y\in V(G)|xy\in E(G)\}$. The degree of a vertex $\displaystyle x$ is the cardinality of $\displaystyle N(x)$ and it is denoted by $\displaystyle d(x)$. We denote by $\displaystyle\Delta(G)$ the maximum degree of $\displaystyle G$. If $\displaystyle x,y\in V(G)$, the distance between $\displaystyle x$ and $\displaystyle y$ (that is, the length of the shortest $\displaystyle x-y$-path) is represented by $\displaystyle d(x,y)$ and $\displaystyle N^{2}(x)=\{y\in V(G)|d(x,y)=2\}$. The girth of $\displaystyle G$ is the length of its shortest cycle, and is denoted by $\displaystyle g(G)$.

Let $\displaystyle G$ be a graph with a proper vertex coloring. Let us denote by $\displaystyle C_{i}$ the set of vertices of color $\displaystyle i$, herein called the class of color $\displaystyle i$. Let $\displaystyle x_{i}$ denote a vertex $\displaystyle x$ of color $\displaystyle i$; $\displaystyle x_{i}$ is said a color-dominating vertex (or, b-dominating vertex) if $\displaystyle x_{i}$ is adjacent to at least one vertex in each of the other classes. A color $\displaystyle i$ is a dominating color if there is at least one vertex $\displaystyle x_{i}$ that is color-dominating. If $\displaystyle y_{i}$ is a vertex which is not color-dominating, at least one color $\displaystyle j\neq i$ does not appear in $\displaystyle N(y_{i})$. The color $\displaystyle j$ is said a missing color in $\displaystyle N(y_{i})$ or simply a missing color of $\displaystyle y_{i}$.

A b-coloring is a proper coloring of its vertices such that each color class contains a color-dominating vertex.

The b-chromatic number $\displaystyle b(G)$ is the largest integer $\displaystyle k$ such that $\displaystyle G$ admits a b-coloring with $\displaystyle k$ colors. Since this parameter has been introduced by R. W. Irving and D. F. Manlove [6], it aroused the interest of many researchers as we can see in [5], [2] and [3] and, more recently, in [7].

For a vertex $\displaystyle x$ of $\displaystyle G$, let $\displaystyle G-\{x\}$ be the vertex-deleted subgraph of $\displaystyle G$ obtained by deleting $\displaystyle x$ and all edges incident to $\displaystyle x$. It is known that the chromatic number of $\displaystyle G-\{x\}$ can have a maximum variation of one unit compared to the chromatic number of $\displaystyle G$. However, this is not true for the b-chromatic number - the difference between $\displaystyle b(G)$ and $\displaystyle b(G-\{x\})$ can be arbitrarily large. This fact motivates the search for bounds to the b-chromatic number (see [1] and [8]). For general graphs S.F.Raj and R.Balakrishnan proved that:

Theorem 1

[1]** For any connected graph of order $\displaystyle n\geq 5$, and for any vertex $\displaystyle x\in V(G)$,

$\displaystyle b(G)-(\lceil n/2\rceil-2)\leq b(G-\{x\})\leq b(G)+(\lfloor n/2\rfloor-2)$**

The bounds are sharp.

In [9], some upper bounds of $\displaystyle b(G-\{x\})$ have been established in some classes of graphs as quasi-line graphs, graphs of large girth and chordal graphs. A chordal graph is a graph such that every cycle of length at least $\displaystyle 4$ has a chord. A graph is a quasi-line graph if the neighborhood of each vertex is covered by at most two cliques. In particular, claw-free graphs (i.e. graphs without induced $\displaystyle K_{1,3}$) are quasi-line graphs. The following results have been shown.

Theorem 2

[9]** Let $\displaystyle G=(V,E)$ a graph.

1) If $\displaystyle G$ is a quasi-line-graph, then for each vertex $\displaystyle x$,

[TABLE]

2) If $\displaystyle G$ is any graph of girth at least $\displaystyle 5$, then for each vertex $\displaystyle x$,

[TABLE]

Theorem 3

[9]** Let $\displaystyle G=(V,E)$ be a chordal graph of clique-number $\displaystyle\omega$ and b-chromatic number $\displaystyle b(G)$. Then, for each vertex $\displaystyle x$,

[TABLE]

[TABLE]

In this work we present lower bounds for $\displaystyle b(G-\{x\})$ in terms of $\displaystyle b(G)$, particularly regarding two important classes of graphs, quasi-line and chordal graphs. We also obtain a lower bound for $\displaystyle b(G-\{x\})$ for graphs of large girth.

Besides this introduction we have three more sections. In the second one we present a lower bound for $\displaystyle b(G-\{x\})$, when $\displaystyle G$ is a general graph and another bound for quasi-line graphs. The third section is devoted to the study of chordal graphs, obtaining also a lower bound for $\displaystyle b(G-\{x\})$ in this class. Finally, in the last section, we analyse graphs with girth at least $\displaystyle 5$, presenting also here a lower bound for $\displaystyle b(G-\{x\})$.

2 General bound and quasi-line graphs

We begin this section with a lower bound for the b-chromatic number of a vertex deleted subgraph of any graph.

Proposition 1

For every vertex $\displaystyle x\in V(G)$, $\displaystyle b(G-\{x\})\geq b(G)-d(x).$

Proof. Let $\displaystyle x\in V(G)$ be a fixed vertex. For a b-coloring of $\displaystyle G$, let $\displaystyle i$ be the color of $\displaystyle x$. We consider two cases. First suppose that each color has at least one color-dominating vertex in $\displaystyle G-\{x\}$; then the b-coloring of $\displaystyle G$ is also a b-coloring of $\displaystyle G-\{x\}$, so $\displaystyle b(G-\{x\})\geq b(G)$. Now, let us consider that there is a color with no color-dominating vertices in $\displaystyle G-\{x\}$. We have then two possibilities:

• •

There is no color-dominating vertex of color $\displaystyle i$ in $\displaystyle G-\{x\}$, that is, $\displaystyle C_{i}$ has no color-dominating vertex then; for each vertex $\displaystyle z$ in $\displaystyle C_{i}$, there is at least one color missing in $\displaystyle N(z)$. We can change the color of each vertex $\displaystyle z$ in $\displaystyle C_{i}$ by a missing color in $\displaystyle N(z)$, eliminating the color $\displaystyle i$.As $\displaystyle C_{i}$ is a stable set, the new coloring is proper. For this case we have $\displaystyle b(G-\{x\})\geq b(G)-1\geq b(G)-d$.

• •

There is a vertex $\displaystyle y\in N(x)$ such that $\displaystyle y$ was the color-dominating vertex of color $\displaystyle s,s\neq i$ in $\displaystyle G$ and there is no more color-dominating vertex of color $\displaystyle s$ in $\displaystyle G-\{x\}$. As $\displaystyle C_{s}$ has no color-dominating vertices, we then change the color $\displaystyle s$ of $\displaystyle y$ by $\displaystyle i$ and, for each other vertex $\displaystyle z$ in $\displaystyle C_{s}$, we change the color $\displaystyle s$ for a missing color of $\displaystyle z$, eliminating the color $\displaystyle s$. As $\displaystyle C_{s}$ is a stable set, the new coloring is proper. We repeat this process for all vertices in $\displaystyle N(x)$ in the same conditions as $\displaystyle y$. We do this for at most $\displaystyle d(x)$ vertices. In this case we obtain $\displaystyle b(G-\{x\})\geq b(G)-d(x)$.

$\displaystyle\blacksquare$

If $\displaystyle\Delta(G)<\left\lceil\frac{n}{2}\right\rceil-2$, this bound is better than the lower bound in [1].

Note that there exist chordal (resp. quasi-line) graphs $\displaystyle G$ such that $\displaystyle b(G-\{x\})$ is stricly less than $\displaystyle b(G)$. For example, let $\displaystyle G_{0}$ be a chordal graph obtained from a chordal graph $\displaystyle H$ and a new vertex $\displaystyle x$ joined to every vertex of $\displaystyle H$. Then $\displaystyle b(G_{0}~{}-x)=b(G_{0})-1$.

Theorem 4

If $\displaystyle G$ is a quasi-line graph then, for every vertex $\displaystyle x\in V(G)$, $\displaystyle b(G-\{x\})\geq b(G)-2$.

Proof. Let $\displaystyle x\in V(G)$ be a fixed vertex. $\displaystyle N(x)$ is covered by at most two cliques $\displaystyle K_{1}$ and $\displaystyle K_{2}$. Let $\displaystyle i$ be the color of $\displaystyle x$.

Considering a b-coloring of $\displaystyle G$, there is at most two vertices $\displaystyle u_{k},u^{\prime}_{k}\in N(x)$ with the same color $\displaystyle k$. Again, by the fact that $\displaystyle G$ is quasi-line, $\displaystyle N^{2}(x)\cap N(u_{k})$ is a clique as it is independent from the neighbour $\displaystyle x$ of $\displaystyle u_{k}$. Analogously $\displaystyle N^{2}(x)\cap N(u^{\prime}_{k})$ is a clique.

We delete the vertex $\displaystyle x$.If in $\displaystyle G-\{x\}$ each color is dominating, then $\displaystyle b(G-\{x\})\geq b(G).$ Let $\displaystyle i$ be the color of $\displaystyle x$in $\displaystyle G$ . We may suppose that $\displaystyle G-\{x\}$ has a color-dominating vertex of color $\displaystyle i$ otherwise we color each vertex of color $\displaystyle i$ by a missing color and we get a b-coloring of $\displaystyle G-\{x\}$ by $\displaystyle b(G)-1$ colors .

We choose a color $\displaystyle s$ that is no more dominating, which means that there is no more color-dominating vertices of color $\displaystyle s$. Each vertex of color $\displaystyle s$ has a missing color.

If the color $\displaystyle s$ had more than one color-dominating vertex in $\displaystyle G$, then it had exactly two color-dominating vertices $\displaystyle w_{s}\in K_{1}$ and $\displaystyle w^{\prime}_{s}\in K_{2}$. We recolor both of them by $\displaystyle i$. We then recolor each other vertex of color $\displaystyle s$ by a missing color. In this way we obtain a b-coloration of $\displaystyle G-\{x\}$, eliminating one color.

If in $\displaystyle G$,there was only one color-dominating vertex $\displaystyle w_{s}$ of color $\displaystyle s$ in $\displaystyle N(x)$, say in $\displaystyle K_{1}$, we recolor $\displaystyle w_{s}$ by $\displaystyle i$. We eliminate the color $\displaystyle s$ by coloring each vertex of $\displaystyle C_{s}$ by a missing color. If there is a color $\displaystyle t$ no more dominating we choose one, then the color-dominating vertex $\displaystyle w_{t}$ was necessarily in $\displaystyle K_{2}$. We color $\displaystyle w_{t}$ by $\displaystyle i$. We color any other vertex of $\displaystyle C_{t}$ by a missing color. Necessarily all the remaining colors are dominating. We conclude that $\displaystyle b(G-\{x\})\geq b(G)-2$.

$\displaystyle\blacksquare$

Note that there exists a quasi-line graph $\displaystyle G$ such that $\displaystyle b(G-x)=b(G)-1$ for at least a vertex $\displaystyle x$.We give an example. Let $\displaystyle\omega\geq 3$ be an integer, and let $\displaystyle p=2\omega-1$. Let $\displaystyle P=\{x_{0},x_{1},...,x_{p},x_{p+1}~{}\}$ be a path. We consider the graph $\displaystyle G_{1}$ obtained by replacing each edge $\displaystyle[x_{i},x_{i+1}]$ by a clique $\displaystyle K_{i}$ of order $\displaystyle\omega$. The graph $\displaystyle G_{1}$ is a claw-free graph; we have $\displaystyle b(G_{1})=p$ and $\displaystyle b(G_{1}-\{x\})=b(G_{1})-1$.

3 Chordal graphs

We want to show the following result.

Theorem 5

Let $\displaystyle G$ be a chordal graph and $\displaystyle x$ be a fixed vertex of $\displaystyle G$. Then $\displaystyle b(G-\{x\})\geq b(G)-\omega_{G}$ where $\displaystyle\omega_{G}$ is the clique number of $\displaystyle G$.

We will need first the next lemma, about the adjacencies in chordal graphs.

Lemma 1

Let $\displaystyle G$ be a chordal graph, and $\displaystyle a,x,b,$ be three consecutive vertices of a cycle $\displaystyle\Gamma$ of $\displaystyle G$. Suppose that the vertex $\displaystyle x$ of $\displaystyle G$ has no neighbours in $\displaystyle\Gamma-\{a,b\}$. Then $\displaystyle a$ and $\displaystyle b$ are adjacent in $\displaystyle G$.

Proof. The proof is by contradiction. We suppose $\displaystyle a$ and $\displaystyle b$ independent. Suppose $\displaystyle\Gamma$ is a shortest cycle containing the path $\displaystyle axb$. If the length of $\displaystyle\Gamma$ is at least $\displaystyle 4$, then as $\displaystyle G$ is chordal, and by minimality of $\displaystyle\Gamma$, it contains a chord incident with $\displaystyle x$ whose second endvertex is distinct from $\displaystyle a$ and $\displaystyle b$, a contradiction.

$\displaystyle\blacksquare$

In what follows, consider a $\displaystyle b$-coloring of $\displaystyle G$, with $\displaystyle b=b(G)$ colors. Let $\displaystyle x\in V(G)$ be a fixed vertex and let $\displaystyle i$ be the color of $\displaystyle x$. Let $\displaystyle I_{a}$ be the set of colors without color-dominating vertices in $\displaystyle G-\{x\}$ and let $\displaystyle J_{a}$ be their set of color-dominating vertices in $\displaystyle G$. We remark that $\displaystyle J_{a}\subset N(x)$ and no vertex in $\displaystyle J_{a}$ is neighbour of a vertex of color $\displaystyle i$ in $\displaystyle G-\{x\}$.

Before proving our main result, we introduce a necessary definition.

Definition 1

Let $\displaystyle x_{i}^{\prime}$ be a fixed color-dominating vertex of color $\displaystyle i$, different from $\displaystyle x$. Let $\displaystyle W=\{w\mid w\notin C_{i},w~{}\mbox{color-dominating vertex in}~{}G\}$. Let $\displaystyle k\leq b$ be a fixed integer. We denote by $\displaystyle W_{k}$ the set of color-dominating vertices of color $\displaystyle k$. A path $\displaystyle P$ of $\displaystyle G-\{x\}$ is said a pseudo-alternating path of $\displaystyle G-\{x\}$, and denoted by $\displaystyle P_{k}[x^{\prime}_{i},z]$, if it is a path of endvertices $\displaystyle x^{\prime}_{i}$ and $\displaystyle z$, such that:

• •

$\displaystyle V(P_{k})\subset C_{i}\cup C_{k}\cup W\cup\{z\}$**

• •

each $\displaystyle w\in W\cap V(P_{k})$ , $\displaystyle w\neq x^{\prime}_{i}$ and $\displaystyle w\neq z$, is preceded by a vertex of color $\displaystyle C_{k}$ (resp. of $\displaystyle C_{i}$) and succeded by a vertex of $\displaystyle C_{i}$ (resp. of $\displaystyle C_{k}$).

• •

$\displaystyle V(P_{k}-\{z\})\cap N(x)=\varnothing$.

A pseudo-alternating path $\displaystyle P[x_{i}^{\prime},z]$ is an alternating path if $\displaystyle z$ is neighbour of $\displaystyle x$ in $\displaystyle G$. We remark that if $\displaystyle z\notin C_{i}\cup C_{k}$, $\displaystyle z$ is necessarily preceded by a vertex of $\displaystyle C_{i}\cup C_{k}$ and, if $\displaystyle P$ is maximal, $\displaystyle z$ has neighbours in $\displaystyle C_{i}$ and $\displaystyle C_{k}$, belonging to $\displaystyle P$ or $\displaystyle z\in I_{a}.$

Proof of the Theorem 5

We consider a $\displaystyle b$-coloring of $\displaystyle G$, with $\displaystyle b(G)$ colors. Suppose $\displaystyle x\in C_{i}$. We may assume that:

There is no $\displaystyle b$-coloring of $\displaystyle G-\{x\}$ by $\displaystyle b(G)-1$ colors      **(a)

**otherwise we have the inequality of the theorem. If $\displaystyle C_{i}-\{x\}$ contains no color-dominating vertex, we recolor each vertex of that set by a missing color in its neighborhood. We get a $\displaystyle b$-coloring of $\displaystyle G-\{x\}$ by $\displaystyle b(G)-1$ colors, a contradiction with assumption (a).

We may suppose from now that $\displaystyle C_{i}-\{x\}$ has color-dominating vertices. $\displaystyle x_{i}^{\prime},x_{i}",\ldots,x_{i}^{(r)}$. Let us take $\displaystyle k$ in $\displaystyle I_{a}$. We recolor each vertex of $\displaystyle C_{k}$ by a missing color. We eliminate color $\displaystyle k$. In this new coloring, if there is a color $\displaystyle k^{\prime}$ with no color-dominating vertex, then $\displaystyle k^{\prime}\in I_{a}$. We recolor $\displaystyle C_{k^{\prime}}$ and we eliminate color $\displaystyle k^{\prime}$. Repeating this process, we get finally a $\displaystyle b$-coloring by at least $\displaystyle b(G)-|I_{a}|$. In view to establish the bound of the theorem, we want to bound $\displaystyle|I_{a}|$. The bound will be established by three claims.

We denote by $\displaystyle Q[y,v]$ any path of $\displaystyle G-\{x\}$ such that $\displaystyle V(Q)\cap N(x)=\{v\}$. Let $\displaystyle v\bar{~{}}$ be the neighbour of $\displaystyle v$ in that path. Let $\displaystyle x_{i}$ be a color-dominating vertex.

Let $\displaystyle F(x_{i})$ be the set of neighbours $\displaystyle z$ of $\displaystyle x$ such that $\displaystyle z$ is extremity of an alternating path $\displaystyle P_{k}[x_{i},z]$ and $\displaystyle k$ is the color of $\displaystyle z$ i.e., $\displaystyle F(x_{i})~{}=F_{1}(x_{i})\cup F_{2}(x_{i})$, where $\displaystyle F_{1}(x_{i})$ and $\displaystyle F_{2}(x_{i})$ are defined below:

$\displaystyle F_{1}(x_{i})=\{z\in N(x)|z=z_{k}~{}\mbox{ and in some}P_{k}[x_{i},z_{k}],{z_{k}}\bar{~{}}~{}\mbox{in}C_{i}~{}\}$

and

$\displaystyle F_{2}(x_{i})=\{z\in N(x)|z=z_{k}~{}\mbox{and in some}P_{k}[x_{i}^{\prime},z_{k}],~{}{z_{k}}\bar{~{}}~{}\mbox{in~{} W}\}\diagdown F_{1}(x_{i}^{\prime})$.

Let $\displaystyle\cal{G}$ be a component of $\displaystyle G-(\{x\}\cup N(x))$ Consider $\displaystyle X_{i}$ the set of color-dominating vertices of color $\displaystyle i$ contained in $\displaystyle\cal{G}$.

Let $\displaystyle F^{\prime}(x_{i})=\{v\in N(x)|,\mbox{there exists~{}}Q[x_{i},v]\}$.

Let $\displaystyle F^{\prime}=\{v\in N(x)|,\mbox{there exists~{}}x_{i}~{}\rm{in}~{}X_{i}~{}\rm{and}~{}Q[x_{i},v]\}$. Note that for any $\displaystyle x_{i}$, we have $\displaystyle F^{\prime}(x_{i})=F^{\prime}=N(x)\cap N(\cal{G}).$

Let $\displaystyle F_{1}=\cup\{F_{1}(x_{i}^{r}),x_{i}^{r}\in\cal{G}\}$.It is a subset of $\displaystyle F^{\prime}.$

By assumption (a) there exists a component $\displaystyle\cal{G}$ for which there is no recoloring of $\displaystyle\cal{G}$ by $\displaystyle\{1,...,b_{G}\}-\{i\}$ such that all the colors of $\displaystyle W\cap({\cal{G}}~{}\cup N(x))$ have a color-dominating vertex in $\displaystyle G-x.$ From now we use such a component.

We get the following assertion as corollary of Lemma 1.

**Claim 1: **For vertex $\displaystyle x_{i}$ of $\displaystyle X_{i}$, $\displaystyle F^{\prime}(x_{i}^{\prime})$ is a clique containing $\displaystyle F_{1}$.

Proof of claim 1: It is sufficient to note that $\displaystyle F^{\prime}(x_{i})$ is not empty, otherwise there is no alternating path, we choose $\displaystyle k\neq i$ and we exchange colors $\displaystyle k$ and $\displaystyle i$ in the pseudo-alternating paths. No color-dominating vertex loses a color. $\displaystyle X_{i}=\{x_{i}^{\prime},\ldots,x_{i}^{r}\}$ is recolored by $\displaystyle k$. A contradiction with the definition of $\displaystyle\cal G$.

$\displaystyle F^{\prime}(x_{i})$ is a clique by Lemma 1 $\displaystyle\bullet$

At this moment we need to introduce another definition

Let $\displaystyle P_{s}[x_{i}^{\prime},z_{1}]$ be an alternating path for $\displaystyle s\neq i$, $\displaystyle s\neq 1$, with $\displaystyle z_{1}\in F_{1}(x_{i}^{\prime})$. An extension of $\displaystyle P_{s}$, denoted by $\displaystyle R_{s}[x_{i}^{\prime},z^{\prime}]$, is a path of the form $\displaystyle P_{s}[x_{i}^{\prime},z_{1}]\cup[z_{1},y_{i}]\cup L(y_{i},z^{\prime}]$, where

• •

$\displaystyle V(L)\subset C_{i}\cup C_{s}\cup W$; $\displaystyle(V(L)-\{z^{\prime}\})\cap N(x)\subset W$.

• •

For each $\displaystyle z\in V(L)-\{z^{\prime}\}$ ,

• •

if $\displaystyle z=z_{k}\in N(x)$, then $\displaystyle z\in W$ a color-dominating vertex of color $\displaystyle k$ and $\displaystyle W_{k}\subset N(x)$; $\displaystyle z_{k}$ is preceded in $\displaystyle R_{s}$ by a vertex of $\displaystyle C_{s}$, followed by a vertex $\displaystyle y_{i}(k)$. If $\displaystyle z\in F_{1}(x_{i}^{l})$, then $\displaystyle y_{i}(k)\in P_{k}(x_{i}^{l},z_{k})$

• •

if $\displaystyle z\in W-N(x)$ then $\displaystyle z$ is preceded by a vertex of $\displaystyle C_{r}$, followed by a vertex of $\displaystyle C_{r}^{\prime}$, where $\displaystyle\{r,r^{\prime}\}=\{i,s\}$

• •

If $\displaystyle z^{\prime}\in W$, $\displaystyle z^{\prime}$ is preceded by a vertex of $\displaystyle C_{i}\cup C_{s}$.

And now, we have two claims:

Let $\displaystyle\cal K~{}(A)$ be the set of colors which appear in the subset $\displaystyle A$ of $\displaystyle V$.

**Claim 2: ** Let $\displaystyle P_{s}[x_{i}^{\prime},z^{\prime}]$ be an alternating extended path. Then $\displaystyle V(L)\cap N(x)$ is a subset of $\displaystyle F_{1}$. So if $\displaystyle z^{\prime}\in W$ then $\displaystyle z^{\prime}\notin J_{a}.$ If $\displaystyle s\notin\cal{K}(F^{\prime})$, then $\displaystyle z^{\prime}\in W~{}-~{}J_{a}$.

Proof of claim 2:

Let $\displaystyle z_{1},z_{2},...,z_{t},..,z_{p}$ be the successive vertices of $\displaystyle V(R_{s})\cap N(x)$ and $\displaystyle z^{\prime}=z_{p}.$ We show by induction on $\displaystyle t$, that $\displaystyle z_{t}$ is in $\displaystyle F_{1}$. Suppose that $\displaystyle z_{t-1}\in F_{1}(x_{i}^{l})$ for some $\displaystyle l$. So there is a path $\displaystyle P_{t-1}[x_{i}^{l},y_{i}^{(t-1)}]$. Composing it with $\displaystyle R_{s}(y_{r}^{(t-1)},z_{t})$, we get a path $\displaystyle Q[x^{(l)},z_{t}]$. If $\displaystyle z_{t}\notin F_{1}$, we do $\displaystyle\tau(i,t)$ from $\displaystyle x_{i}^{l^{\prime}}$ for any $\displaystyle l^{\prime}$, where $\displaystyle\tau(i,t)$ means exchanging the colors $\displaystyle i$ and $\displaystyle t$ in $\displaystyle\cal G$. No color-dominating vertex loses color $\displaystyle t$ even if it is an extremity of an alternating path $\displaystyle P_{t}$ in this later case it is neighbour of $\displaystyle z_{t}$. Some color-dominating neighbours of $\displaystyle N_{t}(x)$ may lose color $\displaystyle i$. We recolor each $\displaystyle v\in X^{\prime}_{i}$ by a missing color. No color-dominating vertex of color $\displaystyle i$ is created by uniqueness of the color-dominating of color $\displaystyle t$. We get a coloring by $\displaystyle b(G)-1$ colors. A contradiction. So $\displaystyle z_{t}\in F_{1}$ and $\displaystyle z_{t}\notin J_{1}$. If $\displaystyle s\notin\cal{K}(F^{\prime})$, then no vertex of $\displaystyle C_{s}$ is in $\displaystyle F_{1}$. So $\displaystyle z^{\prime}\in W$. As $\displaystyle F_{1}\cap J_{a}=\emptyset$, then $\displaystyle z^{\prime}\in W-J_{a}.$

Claim 3: \displaystyle\cal{K}$$\displaystyle(F^{\prime}) contains $\displaystyle I_{a}$

Proof of claim 3:

It is by contradiction. We suppose that there is a color $\displaystyle s$ such that $\displaystyle s\in I_{a}\setminus\cal{K}(F^{\prime})$. Thus no vertex of color $\displaystyle s$ belongs to $\displaystyle F_{1}$.

Case 1: There is no path $\displaystyle P_{s}[x_{i}^{r},z]$ with $\displaystyle x_{i}^{r}\in\cal{G}$ and $\displaystyle z\in N(x)$.

We do $\displaystyle\tau(i,s)$ along the pseudo-alternating paths $\displaystyle P_{s}[x_{i}^{r},u]$, $\displaystyle u\notin N(x)$ simultaneously. No color-dominating vertex loses color. We recolor the remaining vertices of $\displaystyle C_{i}$ in $\displaystyle\cal{G}$ and this leads to a contradiction with the assumption.

Case 2: There exists $\displaystyle x_{i}^{r}$ in $\displaystyle\cal{G}$ and a path $\displaystyle P_{s}[x_{i}^{r},z_{t}]$ with $\displaystyle z_{t}\in N(x)$, $\displaystyle x_{i}^{r}\in\cal{G}$.

This case will be divided in two sub-cases:

Case 2.1: There exists $\displaystyle P_{s}[x_{i}^{r},z_{t}]$ with $\displaystyle z_{t}\notin F_{1}$

By claim 1, $\displaystyle F^{\prime}$ does not contain $\displaystyle z^{\prime}_{t}$, with $\displaystyle z^{\prime}_{t}\neq z_{t}$. We do $\displaystyle\tau(i,t)$ simultaneously in all alternating $\displaystyle P_{t}[x_{i}^{l},z]$, with $\displaystyle x_{i}^{l}$ in $\displaystyle\cal{G}$. As $\displaystyle z_{t}\in F^{\prime}$,by Claim1, the color-dominating vertices contained in $\displaystyle F^{\prime}$ do not lose color $\displaystyle t$, they may lose color $\displaystyle i$. The color-dominating vertices which may lose a color are the color-dominating vertices preterminal in $\displaystyle P_{t}[x_{i}^{l},z_{t}]$; these color-dominating vertices may lose color $\displaystyle i$. We recolor $\displaystyle C_{i}$ by missing colors in $\displaystyle\cal{G}$.

Case 2.2: For any $\displaystyle P_{s}$, the extremity contained in $\displaystyle N(x)$ is in $\displaystyle F_{1}$.

So this extremity does not belong to $\displaystyle J_{a}$.

If for any $\displaystyle x_{i}^{l}\in X_{i}$, there is no alternating path $\displaystyle P_{s}[x_{i}^{l},z]$ with color $\displaystyle s$ preceding $\displaystyle z$, then we do $\displaystyle\tau(i,s)$ in all the pseudo-alternating paths and the paths $\displaystyle P_{s}$. We have the same conclusion as in case 2.1.

If for some $\displaystyle l,z$ is preceded by a vertex of color $\displaystyle s$ in $\displaystyle P_{s}[x_{i}^{l},z]$ we consider the extended paths $\displaystyle R_{s}$. We do $\displaystyle\tau(i,s)$ simultaneously in all alternating and pseudo-alternating paths $\displaystyle P_{s}$ and the extended paths $\displaystyle R_{s}$. The color-dominating vertices which may lose a color are either in $\displaystyle N(x)$ or in $\displaystyle N^{2}(x)$, they are among terminal vertices and preterminal vertices of the alternating paths $\displaystyle P_{s}$ and $\displaystyle R_{s}$; and they may lose color $\displaystyle i$. We then recolor each remaining vertex of $\displaystyle C_{i}\cap{\cal G}$ by a missing color.

In each case we have a contradiction with the definition of the component $\displaystyle\cal G$. So \displaystyle I_{a}\subset\cal K$$\displaystyle(F^{\prime})\bullet

By claim 1 and claim 3 there is a clique containing $\displaystyle x$ and $\displaystyle F^{\prime}$. So we have $\displaystyle\omega(G)\geq|F^{\prime}|+1\geq|I_{a}|+1$, and this finishes the proof of the theorem.

$\displaystyle\blacksquare$

4 Graphs with large girth

The m-degree of a graph $\displaystyle G$, denoted by $\displaystyle m(G)$, is the largest integer $\displaystyle m$ such that $\displaystyle G$ has $\displaystyle m$ vertices of degree at least $\displaystyle m-1$. It is known that, for any graph $\displaystyle G$, $\displaystyle b(G)\leq m(G)$ (see [6]).

Note that A.Campos et al. [4] have shown that graphs of girth at least $\displaystyle 7$ have high b-chromatic number; for each such a graph $\displaystyle G$ this number is at least $\displaystyle m(G)-1$.

We can verify that $\displaystyle m(G-\{x\})\geq m(G)-1.$ Indeed, we have three possibilities to consider: $\displaystyle x$ is one of the $\displaystyle m$ vertices of degree $\displaystyle m-1$; $\displaystyle x$ is a neighbor of one of the $\displaystyle m$ vertices of degree $\displaystyle m-1$, or $\displaystyle x$ is not in any of the previous situations. In the first case, there remain $\displaystyle m-1$ vertices of degree at least $\displaystyle m-2$ and, thus $\displaystyle m(G-\{x\})\geq m(G)-1.$ In the second case, there remain $\displaystyle m$ vertices of degree at least $\displaystyle m-2$, and again, $\displaystyle m(G-\{x\})\geq m(G)-1.$ In the latter case the m-degree does not change, that is, $\displaystyle m(G-\{x\})=m(G)$.

So $\displaystyle b(G-\{x\})\geq b(G)-2$ for graphs of girth at least $\displaystyle 7$. No particular bound is known for graphs of girth $\displaystyle 5$ or $\displaystyle 6$. In this work we show the following.

Theorem 6

*Let $\displaystyle G$ be a graph of girth al least $\displaystyle 5$. For each vertex $\displaystyle x$,

$\displaystyle b(G-\{x\})\geq b(G)-2$*

Proof. Let $\displaystyle\cal B$ be a b-coloring of $\displaystyle G$ and let $\displaystyle i$ be the color of the deleted vertex $\displaystyle x$. Let $\displaystyle W$ be the set of color-dominating vertices of colors different from $\displaystyle i$ in $\displaystyle G$. Let $\displaystyle W_{k}$ be the subset of those of color $\displaystyle k$.

We may suppose that there is a set of color-dominating vertices $\displaystyle X_{i}$ of color $\displaystyle i$, different from $\displaystyle x$. For each vertex $\displaystyle u$ let $\displaystyle K_{1}(u)$ be the set of colors with at least a color-dominating vertex in $\displaystyle N(u)$, let us set $\displaystyle K_{2}(u)=\{1,2,...,b\}-(K_{1}(u)\cup\{i\}).$

We use the notations of the previous section. We may suppose $\displaystyle|I_{1}|\geq 3.$ Note that for $\displaystyle u\neq x$, as the girth is at least $\displaystyle 5$, $\displaystyle K_{1}(u)$ does not contain $\displaystyle I_{1}$. So $\displaystyle K_{2}(u)$ is not empty as it intersects $\displaystyle I_{1}$. By definition of $\displaystyle I_{1}$, the color $\displaystyle i$ is a missing color for each vertex of $\displaystyle J_{1}$ in $\displaystyle G-\{x\}.$ So for any $\displaystyle x^{\prime}\in X_{i}$, $\displaystyle K_{1}(x^{\prime})$ does not intersect $\displaystyle I_{1}$.

(a) Let $\displaystyle x^{\prime}\in X_{i}$. Let $\displaystyle k\in K_{2}(x^{\prime})$ be fixed.

($\displaystyle\bf P_{a}$) If $\displaystyle G-N(N_{k}(x^{\prime}))$ intersects $\displaystyle W_{p}$ for each color $\displaystyle p$ different from $\displaystyle i$,

we color $\displaystyle x^{\prime}$ by $\displaystyle k$, each vertex of $\displaystyle N_{k}(x^{\prime})$ by a missing color. So $\displaystyle|X_{i}|$ decreases. The color-dominating vertices of $\displaystyle K_{1}(x^{\prime})$ may lose color $\displaystyle i$ in their neighborhood.

(b) As long as there exists a vertex $\displaystyle x^{\prime\prime}$ of $\displaystyle X_{i}$ satisfying ($\displaystyle\bf P_{a}$) we do a recoloring.

From now we may suppose that $\displaystyle X_{i}$ is not empty and no vertex of $\displaystyle X_{i}$ satisfies (a).

Lemma 2

Let $\displaystyle x^{\prime}$ be a fixed vertex of $\displaystyle X_{i}$. For each $\displaystyle k\in K_{2}(x^{\prime})$ there exists exactly one $\displaystyle j_{k}$ such that $\displaystyle W_{j_{k}}$ is contained in $\displaystyle N(N_{k}(x^{\prime}))$ and $\displaystyle N(N_{k}(x^{\prime}))\cap W_{j}=\emptyset$ for any other $\displaystyle j$.

Proof. We know that the property $\displaystyle(a)$ is not satisfied. As $\displaystyle g(G)\geq 5$, if a vertex $\displaystyle w_{j}$ is in $\displaystyle N(N_{k}(x^{\prime}))$ then $\displaystyle w_{j}$ is not neighbour of $\displaystyle N_{r}(x^{\prime})$ for $\displaystyle r$ different from $\displaystyle k$. As $\displaystyle(a)$ is not satisfied, it follows that for each $\displaystyle k\in K_{2}(x^{\prime})$, $\displaystyle N_{k}(x^{\prime})$ is neighbour of any vertex of a set $\displaystyle W_{j}$ for some $\displaystyle j$. As $\displaystyle g(G)\geq 5$, $\displaystyle j$ is in $\displaystyle K_{2}(x^{\prime})$ and $\displaystyle W_{j}$ is not neighbour of $\displaystyle N_{t}(x^{\prime})$ for $\displaystyle t\neq k$. It follows that for $\displaystyle k$ fixed, $\displaystyle j$ is unique. This finishes the proof of the lemma.

Let $\displaystyle s$ be a fixed element of $\displaystyle I_{1}$. Let $\displaystyle x^{\prime}\in X_{i}$. We know that the set $\displaystyle I_{1}$, by definition, is a subset of $\displaystyle K_{2}(x^{\prime}).$ By the precedent Lemma there is exactly one $\displaystyle k$ such that $\displaystyle W_{s}\subset N(N_{k}(x^{\prime}))$. We color each vertex of $\displaystyle N_{k}(x^{\prime})$ by a missing color and $\displaystyle x^{\prime}$ by $\displaystyle k$. If $\displaystyle N_{k}(x^{\prime})$ meets $\displaystyle N_{k}(x^{\prime\prime})$ for some $\displaystyle x^{\prime\prime}\in X_{i}$, by the precedent Lemma, we have $\displaystyle W_{s}\subset N(N_{k}(x^{\prime\prime}))$; we color $\displaystyle x^{\prime\prime}$ by $\displaystyle k$ as well and we recolor $\displaystyle N_{k}(x^{\prime\prime})$ by missing colors different from $\displaystyle i$. We recolor so each vertex of $\displaystyle X_{i}$. Then we recolor each vertex of color $\displaystyle i$ by a missing color. If,finally,the color $\displaystyle s$ has no vertex dominating the colors $\displaystyle\{1,..,b\}-\{i\}$, we recolor each vertex $\displaystyle u_{s}$ by a missing color different from $\displaystyle i$.

After this recoloring of color $\displaystyle i$ and eventually color $\displaystyle s$, we get a b-coloring of $\displaystyle G$ by at least $\displaystyle b(G)-2$ colors.

$\displaystyle\blacksquare$

The first author acknowledges partial support by CAPES and CNPq.