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

Abstract

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 G₁,…,G_s such that G_i is strictly $\text{f}{}_{\mathrm{i}}$-degenerate, given nonnegative-integer-valued functions $\text{f}{}_1\text{,…,}\text{f}{}_{\mathrm{s}}$ whose sum is bounded below at each vertex by the degree of that vertex.