Maximum number of colorings of (2k, k2)-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) : G ∈ F2k,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 →∞.