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.