2016/03/23 by Fox, Jacob, Wei, Fan
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.1603.07056
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) = ct 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 32t/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(32t/3-x/3+o(t)n,2t+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.