Counting cliques in graphs with excluded minors

This thesis explores Turán-type extremal problems in graphs that exclude certain minors, focusing on the maximum number of k-cliques such graphs can contain. The first part of the thesis studies planar graphs, which forbid K5 and K3,3 as minors. We determine the maximum number of edges is in a planar graph that contains no cycle of length k, and establish a general upper bound for the number of edges in a planar graph avoiding Ck for any k≥11. The second part addresses the maximum number of k-cliques in Kt-minor-free graphs. We show essentially sharp bounds on the maximum possible number of cliques of order k in a Kt-minor-free graph on n vertices. More precisely, we determine a function C(k,t) such that for each k<t with t−k>>log2 t, every Kt-minor-free graph on n vertices has at most n⋅C(k,t)1+o(1) cliques of order k. We also show that this bound is sharp by constructing a Kt-minor-free graph on n vertices with C(k,t)n cliques of order k. This result answers a question of Wood and Fox–Wei asymptotically up to an ot(1) factor in the exponent, except in the extreme case where k is very close to t.

Date

2025-07-23

Resource Type

Text

Resource Subtype

Dissertation

Full item page