Improved bounds on the b-chromatic number using the independence and chromatic numbers

Abstract

A b-coloring of a graph $G$ is a proper vertex coloring where each color class contains at least one vertex (a b-vertex) adjacent to a vertex in every other color class. The maximum number of colors in such a coloring is the b-chromatic number, $\mathrm{b}(G)$. A ${\mathrm{b}}^{\ast }$-coloring is a variation in which a b-vertex is adjacent to a b-vertex in every other color class. We employ the ${\mathrm{b}}^{\ast }$-coloring to prove that any $n$-vertex graph $G$ with independence number at most $t$ satisfies $\mathrm{b}(G)\le [(t-1)n+t\chi (G)]/(2t-1)$. This bound extends the bounds of Kouider and Zaker (2006) and Alkhateeb and Kohl (2011) and improves the bound in terms of the clique partition number. We show that this bound is sharp for all $t\ge 2$ and $\chi (G)\ge 3$. Furthermore, we provide a refined bound based on the maximum number of vertex-disjoint independent sets of size $t$. Finally, we prove ${\mathrm{b}}^{\ast }(G)\le [(t-2)n+(t-1)\chi (G)]/(2t-3)$ for all $K_{1,t}$-free graphs $G$, a significant improvement over the analogous bound for $\mathrm{b}(G)$.

14 pages, 3 figures