Tree-chromatic number

Abstract

Let us say a graph G has “tree-chromatic number” at most k if it admits a tree-decomposition (T, (X-t : t is an element of V (T))) such that G[X-t] has chromatic number at most k for each t is an element of V (T). This seems to be a new concept, and this paper is a collection of observations on the topic. In particular we show that there are graphs with tree-chromatic number two and with arbitrarily large chromatic number; and for all l >= 4, every graph with no triangle and with no induced cycle of length more than E has tree-chromatic number at most l - 2. (C) 2015 Elsevier Inc. All rights reserved.