Degenerate and star colorings of graphs on surfaces

TL;DR

This paper investigates the coloring properties of graphs on surfaces, establishing bounds on the number of colors needed for degenerate and star colorings based on graph genus, with implications for graph theory and surface embeddings.

Contribution

It proves new bounds on degenerate star colorings for graphs on surfaces, strengthening previous results and demonstrating sharpness up to logarithmic factors.

Findings

Graphs of maximum degree Δ can be colored with O(Δ^{3/2}) colors.

Graphs of genus g can be colored with O(g^{3/5}) colors.

Results are sharp up to a logarithmic factor.

Abstract

We study the degenerate, the star and the degenerate star chromatic numbers and their relation to the genus of graphs. As a tool we prove the following strengthening of a result of Fertin et al.: If $G$ is a graph of maximum degree $\Delta$, then $G$ admits a degenerate star coloring using $O(\Delta^{3/2})$ colors. We use this result to prove that every graph of genus $g$ admits a degenerate star coloring with $O(g^{3/5})$ colors. It is also shown that these results are sharp up to a logarithmic factor.