Odd list-coloring of graphs of small Euler genus with no short cycles of specific types

TL;DR

This paper proves that certain graphs embeddable in the torus or Klein bottle with specific cycle restrictions are odd 5-choosable, establishing optimal bounds for list-coloring in these surfaces.

Contribution

It introduces new results on odd list-coloring for graphs on small Euler genus surfaces with forbidden short cycles, extending known coloring bounds.

Findings

Graphs embeddable in the torus or Klein bottle with no 3, 4, 6 cycles are odd 5-choosable.

Graphs with no 3, 4, 6, and 8 cycles require at most 5 colors for odd list-coloring.

The bounds on the number of colors are proven to be optimal.

Abstract

Odd coloring is a variant of proper coloring and has received wide attention. We study the list-coloring version of this notion in this paper. We prove that if $G$ is a graph embeddable in the torus or the Klein bottle with no cycle of length 3, 4, and 6 such that no 5-cycles share an edge, then for every function $L$ that assigns each vertex of $G$ a set $L(v)$ of size 5, there exists a proper coloring that assigns each vertex $v$ of $G$ an element of $L(v)$ such that for every non-isolated vertex, some color appears an odd number of times on its neighborhood. In particular, every graph embeddable in the torus or the Klein bottle with no cycle of length 3, 4, 6, and 8 is odd 5-choosable. The number of colors in these results are optimal, and there exist graphs embeddable in those surfaces of girth 6 requiring six or seven colors.