Counting Colorings of a Regular Graph

AbstractAt 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 for each q ≥  3 the number of proper q-colorings admitted by an n-vertex, d-regular graph is at most $$\begin{array}{ll}(q^2/4)^\frac{n}{2}\binom{q}{q/2}^\frac{n(1+o(1))}{2d} &{\rm if}\; q \; {\rm is}\; {\rm even}\\((q^2-1)/4)^\frac{n}{2}\binom{q+1}{(q+1)/2}^\frac{n(1+o(1))}{2d} & {\rm if}\; q\; {\rm is}\; {\rm odd,}\end{array}$$(q2/4)n2qq/2n(1+o(1))2difqiseven((q2-1)/4)n2q+1(q+1)/2n(1+o(1))2difqisodd,where $${o(1)\rightarrow 0}$$o(1)→0 as $${d \rightarrow \infty}$$d→∞ ; these bounds agree up to the o(1) terms with the counts of q-colorings of n/2d copies of Kd,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 $${\delta=\delta(\varepsilon,q)}$$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 $${n(1-\varepsilon)/2}$$n(1-ε)/2 is at most $$(q^2/4-\delta)^\frac{n}{2},$$(q2/4-δ)n2,with an analogous result for odd q.