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 2 d | n , none admits more q -colorings than the disjoint union of n /2 d copies of the complete bipartite graph K d , d . In this note we give asymptotic evidence for this conjecture, showing that for each q ≥ 3 the number of proper q -colorings admitted by an n -vertex, d -regular graph is at most ( q 2 / 4 ) n 2 q q / 2 n ( 1 + o ( 1 ) ) 2 d if q is even ( ( q 2 - 1 ) / 4 ) n 2 q + 1 ( q + 1 ) / 2 n ( 1 + o ( 1 ) ) 2 d 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 /2 d copies of K d , d . Along the way we obtain 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 ≥ 4 and fixed ɛ > 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 2 / 4 - δ ) n 2 , with an analogous result for odd q .