TURAN GRAPHS AND THE NUMBER OF COLORINGS

We consider an old problem of Linial and Wilf to determine the structure of graphs which allows the maximum number of q-colorings among graphs with n vertices and m edges. We show that if r divides q, then for all sufficiently large n the Turan graph T-r(n) has more q-colorings than any other graph with the same number of vertices and edges. This partially confirms a conjecture of Lazebnik. Our proof builds on methods of Loh, Pikhurko, and Sudakov, which reduces the problem to a quadratic program.