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 $0 < \epsilon < \frac{3}{328}$ and for every integer $k\ge \frac{140}{41\epsilon}$, each k-map graph with n vertices has at most $(\frac{325}{328}+\epsilon)kn -- 2k$ edges. 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 $(\frac{11}{12}k+\frac{1}{3})n$ edges.

The full version can be found at http://rnc.r.dendai.ac.jp/~chen/papers/kmap.pdf