A Brooks-like result for graph powers

TL;DR

This paper extends Brooks' theorem to graph powers, showing that for high enough powers, the chromatic number can be significantly lower than the naive upper bound, with precise bounds depending on maximum degree and power.

Contribution

It generalizes Brooks' theorem to graph powers, establishing new bounds on the chromatic number that improve previous results for large powers.

Findings

For $k eq 1$, the chromatic number can be reduced by 2 except for Moore graphs.

For $k eq 1$, the reduction can be improved to $ ext{Theta}(( ext{max degree}-1)^{k/12})$.

The results hold for all sufficiently large maximum degrees and powers.

Abstract

Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic number of $G$, denoted by $\chi(G)$. We investigate the chromatic number of powers of graphs, i.e. the graphs obtained from a graph $G$ by adding an edge between every pair of vertices at distance at most $k$. For $k=1$, Brooks' theorem states that every connected graph of maximum degree $\Delta\geqslant 3$ excepted the clique on $\Delta+1$ vertices can be colored using $\Delta$ colors (i.e. one color less than the naive upper bound). For $k\geqslant 2$, a similar result holds: excepted for Moore graphs, the naive upper bound can be lowered by 2. We prove that for $k\geqslant 3$ and for every $\Delta$, we can actually spare $k-2$ colors, excepted for a finite number of graphs. We then improve this value to $\Theta((\Delta-1)^{\frac{k}{12}})$.