On Oriented Colourings of Graphs on Surfaces

TL;DR

This paper investigates the minimum number of colours needed to properly colour any oriented graph on a surface of genus g, establishing nearly linear bounds that improve previous results.

Contribution

The paper proves that the oriented chromatic number of graphs on surfaces grows nearly linearly with genus, significantly tightening previous bounds and resolving an open question.

Findings

Established nearly linear bounds for hi_o(g) in terms of genus g

Improved previous bounds from polynomial to nearly linear growth

Resolved an open problem posed by the author and colleagues

Abstract

For an oriented graph $G$, the least number of colours required to oriented colour $G$ is called the oriented chromatic number of $G$ and denoted $\chi_o(G)$.For a non-negative integer $g$ let $\chi_o(g)$ be the least integer such that $\chi_o(G) \leq \chi_o(g)$ for every oriented graph $G$ with Euler genus at most $g$. We will prove that $\chi_o(g)$ is nearly linear in the sense that $\Omega(\frac{g}{\log(g)}) \leq \chi_o(g) \leq O(g \log(g))$. This resolves a question of the author, Bradshaw, and Xu, by improving their bounds of the form $\Omega((\frac{g^2}{\log(g)})^{1/3}) \leq \chi_o(g)$ and $\chi_o(g) \leq O(g^{6400})$.