On quas-monotonous graphs

Mekkia Kouider (1)

1 LRI - Laboratoire de Recherche en Informatique

A {\it dominating coloring} by $k$ colors is a proper $k$ coloring where every color $i$ has a representative vertex $x_i$ adjacent to at least one vertex in each of the other classes. The {\it b-chromatic number}, $b(G)$, of a graph $G$ is the largest integer $k$ such that $G$ admits a dominating coloring by $k$ colors.\ A graph $G=(V,E)$ is said {\it $b$-monotonous} if \rm $b(H_1) \geq b(H_2)$ for every induced subgraph $H_1$ of $G$ and every subgraph $H_2$ of $H_1$. Here we say that a graph $G$ is {\it quasi $b$-monotonous }, or simply quasi-monotonous, if for every vertex $v \in V$, $b(G-v) \leq b(G)+1$. We show study the quasi-monotonicity of several classes. We show in particular that chordal graphs are not quasi-monotonous in general,whereas chordal graphs with large b-chromatic number, and $(P,coP,chair,cochair)$-free graphs are quasi-monotonous; $(P_5,coP_5,P)$-free graphs are monotonous.Finally we give new bounds for the b-chromatic number of any vertex deleted subgraph of a chordal graph.

Mots clés: b-chromatic; quasi-monotonous graphs

Domaines: Mathématique discrète [cs.DM]

Mekkia Kouider. On quas-monotonous graphs. 2012. ⟨hal-00801393⟩