New bounds on the edge number of a k-map graph

It is known that for every integer k ≥ 4, each k-map graph with n vertices has at most kn − 2k edges. Previously, it was open whether this bound is tight or not. We show that this bound is tight for k = 4, 5. We also show that this bound is not tight for large enough k (namely, k ≥ 374); more precisely, we show that for every 0 < < 3 328 and for every integer k ≥ 140 41 , each k-map graph with n vertices has at most ( 325 328 + )kn − 2k edges. This result implies the first polynomial (indeed linear) time algorithm for coloring a given k-map graph with less than 2k colors for large enough k. We further show that for every positive multiple k of 6, there are infinitely many integers n such that some k-map graph with n vertices has at least ( 11 12k + 1 3 )n edges.