Colouring powers and girth

TL;DR

This paper investigates the maximum chromatic number of the t-th power of graphs with bounded degree and girth, providing sharp bounds for t ≥ 3 using probabilistic and combinatorial methods.

Contribution

It offers new upper and lower bounds for the chromatic number of graph powers under girth and degree constraints, improving understanding of graph coloring complexities.

Findings

Upper bounds derived using probabilistic methods.

Lower bounds constructed via incidence structures.

Bounds are sharp up to a constant factor as degree increases.

Abstract

Alon and Mohar (2002) posed the following problem: among all graphs $G$ of maximum degree at most $d$ and girth at least $g$, what is the largest possible value of $\chi(G^t)$, the chromatic number of the $t$th power of $G$? For $t\ge 3$, we provide several upper and lower bounds concerning this problem, all of which are sharp up to a constant factor as $d\to \infty$. The upper bounds rely in part on the probabilistic method, while the lower bounds are various direct constructions whose building blocks are incidence structures.