Number of Cliques in Graphs with a Forbidden Subdivision

Abstract

Abstract. We prove that for all positive integers t, every nvertex graph with no K t -subdivision has at most 2 50t n cliques. We also prove that asymptotically, such graphs contain at most 2 (5+o(1))t n cliques, where o(1) tends to zero as t tends to infinity. This strongly answers a question of D. Wood asking if the number of cliques in n-vertex graphs with no K t -minor is at most 2 ct n for some constant c.

Reconfiguration of Cliques in a Graph

“…By Theorem 4 it suffices to show that the claimed graphs have polynomially many maximal cliques. Polynomial bounds on the number of maximal cliques are shown for even-hole-free graphs in [5], for graphs of bounded boxicity in [18], and for K t -subdivision-free graphs in [15].…”

On the number of cliques in graphs with a forbidden minor

“…We assume that G is a graph with n vertices and no K t -minor. Lee and Oum [8] (and in earlier works such as [5]) present a simple algorithm to enumerate all the cliques in a graph G, called the "peeling process". It is helpful in bounding the number of cliques in a graph.…”

“…Norine, Seymour, Thomas, and Wollan [11] and independently Reed and Wood [12] showed that for each t there is a constant c(t) such that every graph on n vertices with no K t -minor has at most c(t)n cliques. Wood [17] asked in 2007 if c(t) < c t for some absolute constant c. Progress on this question was made in [3], and it was recently resolved by Lee and Oum [8]. They proved that every graph on n vertices with no K t -subdivision (and hence every graph on n vertices with no K t -minor) has at most 2 5t+o(t) n cliques, and observed that optimizing their proof would improve the exponential constant from 5 to a number less than 4.…”

Cliques in Graphs Excluding a Complete Graph Minor

“…Upper bounds on cliques(n, t) have been intensely studied over the past ten years, culminating in the recent upper bound by Fox and Wei [14] that matches the lower bound in (5) up to a lower order term. These results are summarised in the following Lee and Oum [23] 2 2(log 2 3)t/3+o(t) n Fox and Wei [14] Note that several authors have also studied the maximum number of cliques in graphs excluding a given subdivision [15,23] or immersion [15].…”

“…This paper considers these questions in graph classes defined by an excluded minor, thus generalising the above results for planar graphs. This direction has been recently pursued by several authors [13,14,23,26,28]. These works have focused on asymptotic results when the excluded minor is a general complete graph K t .…”