A Brooks-like result for graph powers

Coloring a graph G consists in finding an assignment of colors c : V(G) -> {1, ... , 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 increment 3 except the clique on increment + 1 vertices can be colored using increment colors (i.e. one color less than the naive upper bound). For k 2, a similar result holds: except for Moore graphs, the naive upper bound can be lowered by 2. We prove that for k 3 and for every increment , we can actually spare k-2 colors, except for a finite number of graphs. We then improve this value to Theta(( increment - 1)k12). (c) 2023 Elsevier Ltd. All rights reserved.