  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 K_{d,d}. In this note we give asymptotic evidence for this
conjecture, giving an upper bound on the number of proper q-colorings admitted
by an n-vertex, d-regular graph of the form a^n b^{n(1+o(1))/d} (where a and b
depend on q and where o(1) goes to 0 as d goes to infinity) that agrees up to
the o(1) term with the count of q-colorings of n/2d copies of K_{d,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 \epsilon > 0 there is \delta=\delta(\epsilon,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-\epsilon)/2 is at most (a-\delta)^n.
