On the number of cliques in graphs with a forbidden subdivision or immersion

Abstract

How many cliques can a graph on $n$ vertices have with a forbidden substructure? Extremal problems of this sort have been studied for a long time. This paper studies the maximum possible number of cliques in a graph on $n$ vertices with a forbidden clique subdivision or immersion. We prove for $t$ sufficiently large that every graph on $n\ge t$ vertices with no $K_t$-immersion has at most $2^{t+{\log }^{2}t}n$ cliques, which is sharp apart from the $2^{O({\log }^{2}t)}$ factor. We also prove that the maximum number of cliques in an $n$-vertex graph with no $K_t$-subdivision is at most $2^{1.817t}n$. This improves on the best known exponential constant by Lee and Oum. We conjecture that the optimal bound is $3^{2t/3+o(t)}n$, as we proved for minors in place of subdivision in earlier work.

16 pages of main text, 6 pages of appendix

References in corpus (1)

• Rank-width and Tree-width of H-minor-free Graphs