  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
$\chi_t(G)$ 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 $\chi_1$ is the chromatic
number, and $\chi_2$ is known as the point aboricity. The point partition
number $\chi_t$ 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 $\omega_t(G)$ 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 $\omega_t(G)\leq \chi_t(G)$ 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, we give a characterization of
$\chi_t$-perfect graphs of $\mathcal{MG}_t$ by a set of forbidden induced
subgraphs. We also discuss some complexity problems for the class of
$\chi_t$-critical graphs.

Combinatorics