Maximum number of colorings of (2k, k 2 )-graphs

Let ℱ2k, k2 consist of all simple graphs on 2k vertices and k2 edges. For a simple graph G and a positive integer λ, let PG(λ) denote the number of proper vertex colorings of G in at most λ colors, and let f(2k, k2,λ) = max {PG(λ):Geℱ2k,k2}. We prove that f(2k, k2, 3) = PKk,k(3) and Kk,k is the only extremal graph. We also prove that f(2k, k2, 4) = (6+o(1))4k as k ➝ ∞. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 135148, 2007