Subgraph‐avoiding coloring of graphs

Abstract

Abstract Given a “forbidden graph” F and an integer k , an F‐avoiding k‐coloring of a graph G is a k ‐coloring of the vertices of G such that no maximal F ‐free subgraph of G is monochromatic. The F‐avoiding chromatic number ac F ( G ) is the smallest integer k such that G is F ‐avoiding k ‐colorable. In this paper, we will give a complete answer to the following question: for which graph F , does there exist a constant C , depending only on F , such that ac F ( G ) ⩽ C for any graph G ? For those graphs F with unbounded avoiding chromatic number, upper bounds for ac F ( G ) in terms of various invariants of G are also given. Particularly, we prove that \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${{ac}}_{{{F}}}({{G}})\le {{2}}\lceil\sqrt{{{n}}}\rceil+{{1}}$\end{document} , where n is the order of G and F is not K k or \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$\overline{{{K}}_{{{k}}}}$\end{document} . © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 300–310, 2010