On the number of cliques in graphs with a forbidden minor

摘要

Reed and Wood and independently Norine, Seymour, Thomas, and Wollan proved that for each positive integer t there is a constant c(t) such that every graph on n vertices with no Kt-minor has at most c(t)n cliques. Wood asked in 2007 if we can take c(t) = c(t) for some absolute constant c. This question was recently answered affirmatively by Lee and Oum. In this paper, we determine the exponential constant. We prove that every graph on n vertices with no Kt-minor has at most 3(2t)/3+o(t)n cliques. This bound is tight for n >= 4t/3. More generally, let H be a connected graph on t vertices, and x denote the size (i.e., the number edges) of the largest matching in the complement of H. We prove that every graph on n vertices with no H-minor has at most max(3(2t)/(3-)x/3+o(t)(n,2)t+o(t)n) cliques, and this bound is tight for n >= max(4t/3 2x/3, t) by a simple construction. Even more generally, we determine explicitly the exponential constant for the maximum number of cliques an n -vertex graph can have in a minor-closed family of graphs which is closed under disjoint union. (C) 2017 Elsevier Inc. All rights reserved.