The Chromatic Number Of Graph Powers

The square G of a graph G = (V,E) is the graph whose vertex set is V and in which two distinct vertices are adjacent if and only if their distance in G is at most 2. What is the maximum possible chromatic number of G, as G ranges over all graphs with maximum degree d and girth g? Our (somewhat surprising) answer is that for g = 3, 4, 5 or 6 this maximum is (1 + o(1))d (where the o(1) term tends to 0 as d tends to infinity), whereas for all g ≥ 7, this maximum is of order d/ log d. To state this result more precisely, for every two integers d ≥ 2 and g ≥ 3, define f2(d, g) to be the maximum possible value of χ(G) over all graphs with maximum degree d and girth g. Since the maximum degree of G is at most d+ d(d− 1) = d, it follows that for every g