Variable degeneracy: extensions of Brooks' and Gallai's theorems

We introduce the concept of variable degeneracy of a graph extending that of k-degeneracy. This makes it possible to give a common generalization of the point partition numbers and the list chromatic number. In particular, the list point arboricity of a graph is considered. We extend Brooks' and Gallai's theorems in terms of covering the vertices of a graph by disjoint induced subgraphs G(l),...,G(s) such that G(i) is strictly f(i)-degenerate, given nonnegative-integer-valued functions f(l),...,f(s) whose sum is bounded below at each vertex by the degree of that vertex. (C) 2000 Elsevier Science B.V. All rights reserved.