Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. A graph G is said to be 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) is the least integer k for which G admits a coloring with k colors such that each color class induces a strictly t-degenerate subgraph of G. So χ 1 is the chromatic number and χ 2 is the point aboricity. The point partition number χ t with t ≥ 1 was introduced by Lick and White. A graph G is called χ t -critical if every proper subgraph H of G satisfies χ t (H) < χ t (G). In this paper we prove that if G is a χ t -critical graph whose order satisfies |G| ≤ 2χ t (G) − 2, then G can be obtained from two non-empty disjoint subgraphs G 1 and G 2 by adding t edges between any pair u, v of vertices with u ∈ V (G 1 ) and v ∈ V (G 2 ). Based on this result we establish the minimum number of edges possible in a χ t -critical graph G with χ t (G) = k and |G| ≤ 2k − 1 for t even. For t = 1 the corresponding two results were obtained in 1963 by Tibor Gallai.

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Graphs considered in this paper are finite, undirected and without loops, but with multiple edges. For an integer t ≥ 1, denote by 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 ≥ 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 MG t . Then ω t (G) ≤ 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 MG t by a set of forbidden induced subgraphs. We also discuss some complexity problems for the class of χ t -critical graphs.

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