Point Partition Numbers: Perfect Graphs

Abstract

Graphs considered in this paper are finite, undirected and without loops, but with multiple edges. For an integer $$t\ge 1$$ , denote by $$\mathcal{MG}_t$$ the class of graphs whose maximum multiplicity is at most t. A graph G is called strictly t-degenerate if every non-empty subgraph H of G contains a vertex v whose degree in H is at most $$t-1$$ . The point partition number $$\chi _t(G)$$ of G is the smallest number of colors needed to color the vertices of G so that each vertex receives a color and vertices with the same color induce a strictly t-degenerate subgraph of G. So $$\chi _1$$ is the chromatic number, and $$\chi _2$$ is known as the point aboricity. The point partition number $$\chi _t$$ with $$t\ge 1$$ was introduced by Lick and White (Can J Math 22:1082–1096, 1970). If H is a simple graph, then tH denotes the graph obtained from H by replacing each edge of H by t parallel edges. Then $$\omega _t(G)$$ is the largest integer n such that G contains a $$tK_n$$ as a subgraph. Let G be a graph belonging to $$\mathcal{MG}_t$$ . Then $$\omega _t(G)\le \chi _t(G)$$ and we say that G is $$\chi _t$$ -perfect if every induced subgraph H of G satisfies $$\omega _t(H)=\chi _t(H)$$ . Based on the Strong Perfect Graph Theorem due to Chudnowsky, Robertson, Seymour and Thomas (Ann Math 164:51–229, 2006), we give a characterization of $$\chi _t$$ -perfect graphs of $$\mathcal{MG}_t$$ by a set of forbidden induced subgraphs (see Theorems 2 and 3). We also discuss some complexity problems for the class of $$\chi _t$$ -critical graphs.