1 research outputs found
Cover and variable degeneracy
Let be a nonnegative integer valued function on the vertex set of a
graph. A graph is {\bf strictly -degenerate} if each nonempty subgraph
has a vertex such that . In this
paper, we define a new concept, strictly -degenerate transversal, which
generalizes list coloring, signed coloring, DP-coloring, -forested-coloring,
and -partition. A {\bf cover} of a graph is a
graph with vertex set , where ; the edge set , where is a matching between
and . A vertex set is a {\bf transversal} of if
for each . A transversal is a {\bf
strictly -degenerate transversal} if is strictly -degenerate. The
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 -degenerate transversal. Using these results, we can uniformly
prove many new and known results. In the final section, we pose some open
problems