Cover and variable degeneracy

Abstract

Let f be a nonnegative integer valued function on the vertex-set of a graph. A graph is strictly f -degenerate if each nonempty subgraph Γ has a vertex v such that deg Γ (v) < f (v). In this paper, we define a new concept, strictly f -degenerate transversal, which generalizes list coloring, ( f 1 , f 2 , . . . , f κ )-partition, signed coloring, DP-coloring andThe main result of this paper is a degree type result, which generalizes Brooks' theorem, Gallai's theorem, degree-choosable result, signed degree-colorable result, and DP-degree-colorable result. Similar to Borodin, Kostochka and Toft's variable degeneracy, this degree type result is also self-strengthening. We also give some structural results on critical graphs with respect to strictly f -degenerate transversal. Using these results, we can uniformly prove many new and known results. In the final section, we give some open problems.

DP-3-coloring of planar graphs without certain cycles

“…A graph is minimal non-DP-k-colorable if it is not DP-k-colorable but every subgraph with fewer vertices is DP-k-colorable. The following structural results for the minimal non-DP-k-colorable graph are consequences of Theorems in [19].…”