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 12 k + 1 3 )n edges. Key words. Planar graph, plane embedding, sphere embedding, map graph, k-map graph, edge number. AMS subject classifications. 05C99, 68R05, 68R10. Abbreviated title. Edge Number of k-Map Graphs.

• Type: Journal
• Year: 2007
• Where: JGT
• Authors: Zhi-Zhong Chen
• Keywords: Integer, JGT 2007, K-map Graphs, Large Enough