A Brooks-like result for graph powers

Abstract

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 χ(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 Δ⩾3 except the clique on Δ+1 vertices can be colored using Δ 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 Δ, we can actually spare k−2 colors, except for a finite number of graphs. We then improve this value to Θ((Δ−1)k12).