Variable degeneracy: extensions of Brooks’ and Gallaiʹs theorems Original Research Article

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 Gallaiʹs theorems in terms of covering the vertices of a graph by disjoint induced subgraphs G1,…,Gs such that Gi is strictly fi-degenerate, given nonnegative-integer-valued functions f1,…,fs whose sum is bounded below at each vertex by the degree of that vertex.