Point partition numbers: Decomposable and indecomposable critical graphs

Abstract

Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number χt(G) is the least integer k for which G admits a coloring with k colors such that each color class induces a (t−1)-degenerate subgraph of G. So χ1 is the chromatic number and χ2 is the point arboricity. The point partition number χt with t≥1 was introduced by Lick and White. A graph G is called χt-critical if every proper subgraph H of G satisfies χt(H)<χt(G). In this paper we prove that if G is a χt-critical graph whose order satisfies |G|≤2χt(G)−2, then G can be obtained from two non-empty disjoint subgraphs G1 and G2 by adding t edges between any pair u,v of vertices with u∈V(G1) and v∈V(G2). Based on this result we establish the minimum number of edges possible in a χt-critical graph G of order n and with χt(G)=k, provided that n≤2k−1 and t is even. For t=1 the corresponding two results were obtained in 1963 by Tibor Gallai.