The choice number versus the chromatic number for graphs embeddable on orientable surfaces

TL;DR

This paper investigates the relationship between the choice number and chromatic number of graphs on orientable surfaces, establishing bounds on their differences based on surface genus.

Contribution

It provides new bounds on the gap between choice number and chromatic number for graphs on surfaces, especially on the torus and for higher genus surfaces.

Findings

Gap between choice number and chromatic number is at most 2 for certain torus graphs.

Largest gap for graphs on genus g surfaces is proportional to ().

For graphs with low chromatic number relative to (), the gap is o().

Abstract

We show that for loopless $6$-regular triangulations on the torus the gap between the choice number and chromatic number is at most $2$. We also show that the largest gap for graphs embeddable in an orientable surface of genus $g$ is of the order $\Theta(\sqrt{g})$, and moreover for graphs with chromatic number of the order $o(\sqrt{g}/\log_{2}(g))$ the largest gap is of the order $o(\sqrt{g})$.