Complexity of Determining the b-continuity Property of Graphs

le numéro est 2003/37This paper deals with b-colorings of a graph $G$, i.e., proper colorings in which for each color, there exists at least one vertex it is assigned to such that each other color is assigned to at least one of its neighboor. The maximal cardinality of such a $b$-coloring is denoted by $b(G)$, and each proper coloring with cardinal $\xi(G)$ is a $b$-coloring. We say that $G$ is b-continuous iff for each $k$, $\xi(G) \leq k \leq b(G)$, there exists a b-coloring with cardinal $k$. It is well known that no all graphs are $b$-continuous. Calling $b$-spectrum of $G$ the set of cardinals of all the b-colorings of $G$, we first show that for any integer set $I$, there exists a graph which $b$-spectrum is $I$. Then, we show that, even if $b$-coloring s of cardinal $\xi(G)$ and $b(G)$ are given, the problem of knowing if $G$ is $b $-continuous is NP-complete. At end, we show that interval graphs are $b$-continuous