Single-conflict colouring

TL;DR

This paper introduces the concept of single-conflict chromatic number in multigraphs, analyzing its bounds for graphs embeddable on surfaces of given genus, and establishes an asymptotically tight upper bound.

Contribution

It defines the single-conflict chromatic number and provides an asymptotically tight bound for graphs on surfaces of Euler genus g.

Findings

Single-conflict chromatic number is bounded by O(g^{1/4} log g) for graphs on surfaces of genus g.

The bound is sharp up to a logarithmic factor.

The parameter relates closely to separation and adaptable choosability.

Abstract

Given a multigraph, suppose that each vertex is given a local assignment of $k$ colours to its incident edges. We are interested in whether there is a choice of one local colour per vertex such that no edge has both of its local colours chosen. The least $k$ for which this is always possible given any set of local assignments we call the {\em single-conflict chromatic number} of the graph. This parameter is closely related to separation choosability and adaptable choosability. We show that single-conflict chromatic number of simple graphs embeddable on a surface of Euler genus $g$ is $O(g^{1/4}\log g)$ as $g\to\infty$. This is sharp up to the logarithmic factor.