Counting Colorings of a Regular Graph

At most how many (proper) q-colorings does a regular graph admit? Galvin and Tetali conjectured that among all n-vertex, d-regular graphs with 2d|n, none admits more q-colorings than the disjoint union of n/2d copies of the complete bipartite graph Kd,d. In this note we give asymptotic evidence for this conjecture, showing that the number of proper q-colorings admitted by an n-vertex, d-regular graph is at most ( q2/4 )n 2 ( q q/2 )n(1+o(1)) 2d if q is even ( (q2 − 1)/4 )n 2 ( q+1 (q+1)/2 )n(1+o(1)) 2d if q is odd, where o(1) → 0 as d → ∞; these bounds agree up to the o(1) terms with the counts of q-colorings of n/2d copies of Kd,d. An auxiliary result is an upper bound on the number of colorings of a regular graph in terms of its independence number. For example, we show that for all even q and fixed ε > 0 there is δ = δ(ε, q) such that the number of proper q-colorings admitted by an n-vertex, d-regular graph with no independent set of size n(1− ε)/2 is at most ( q/4− δ )n 2 , with an analogous result for odd q.