Colouring powers and girth

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 $χ(G^t)$, 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.