On the strong chromatic number of graphs

Abstract

The strong chromatic number, $\chi_S(G)$, of an $n$-vertex graph $G$ is the smallest number $k$ such that after adding $k\lceil n/k\rceil-n$ isolated vertices to $G$ and considering {\bf any} partition of the vertices of the resulting graph into disjoint subsets $V_1, \ldots, V_{\lceil n/k\rceil}$ of size $k$ each, one can find a proper $k$-vertex-coloring of the graph such that each part $V_i$, $i=1, \ldots, \lceil n/k\rceil$, contains exactly one vertex of each color. For any graph $G$ with maximum degree $\Delta$, it is easy to see that $\chi_S(G)\geq\Delta+1$. Recently, Haxell proved that $\chi_S(G) \leq 3\Delta -1$. In this paper, we improve this bound for graphs with large maximum degree. We show that $\chi_S(G)\leq 2\Delta$ if $\Delta \geq n/6$ and prove that this bound is sharp.