A variation on Heawood list-coloring for graphs on surfaces

TL;DR

This paper extends Heawood's list-coloring theorem to graphs on surfaces with a distinguished face, establishing new list-coloring bounds based on the surface's Euler genus, except for a specific case.

Contribution

It introduces a variation of Heawood list-coloring for embedded graphs with a distinguished face, generalizing Thomassen's planar 5-list-coloring theorem to higher genus surfaces.

Findings

List-coloring is possible under specified list sizes for most surface embeddings.

The induced subgraph on the distinguished face must not contain a complete graph of size H(epsilon)-1.

The result applies to all surfaces except when epsilon equals 3.

Abstract

We prove a variation on Heawood list-coloring for graphs on surfaces, modeled on Thomassen's planar 5-list-coloring theorem. For epsilon>0 define the Heawood number to be H(epsilon)=Floor((7+Sqrt[24*epsilon+1])/2). We prove that, except for epsilon=3, every graph embedded on a surface of Euler genus epsilon>0 with a distinguished face F can be list-colored when the vertices of F have (H(epsilon)-2)-lists and all other vertices have H(epsilon)-lists unless the induced subgraph on the vertices of F contains the complete graph on H(epsilon)-1 vertices.