New bounds on the number of edges in a k-map graph

Abstract

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. We also show that this bound is not tight for large enough k; more precisely, we show that for every 0 < ∈