Let us say a graph G has "tree-chromatic number" at most k if it admits a tree-decomposition (T, (Xt:t∈V(T))) such that G[Xt] has chromatic number at most k for each t∈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 ℓ has tree-chromatic number at most ℓ-2.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Induced subgraph