Point partition numbers: perfect graphs

Abstract

Graphs considered in this paper are finite, undirected and without loops, but with multiple edges. For an integer $t\geq 1$, 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 $χ_t(G)$ of $G$ is 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 $χ_1$ is the chromatic number, and $χ_2$ is known as the point aboricity. The point partition number $χ_t$ with $t\geq 1$ was introduced by Lick and White. 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 $ω_t(G)$ 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 $ω_t(G)\leq χ_t(G)$ and we say that $G$ is $χ_t$-perfect if every induced subgraph $H$ of $G$ satisfies $ω_t(H)=χ_t(H)$. Based on the Strong Perfect Graph Theorem due to Chudnowsky, Robertson, Seymour and Thomas, we give a characterization of $χ_t$-perfect graphs of $\mathcal{MG}_t$ by a set of forbidden induced subgraphs. We also discuss some complexity problems for the class of $χ_t$-critical graphs.