The maximum number of $K_{r_1,\ldots,r_s}$ in graphs with a given circumference or matching number

TL;DR

This paper determines the maximum number of complete multipartite subgraphs in 2-connected graphs with specified circumference or matching number, extending previous results and considering large minimum degree and detour order.

Contribution

It provides new extremal bounds for the number of $K_{r_1, dots,r_s}$ in graphs with given circumference or matching number, including conditions on minimum degree and detour order.

Findings

Maximum number of $K_{r_1, dots,r_s}$ in 2-connected graphs with given circumference.

Maximum number of $K_{r_1, dots,r_s}$ in graphs with given matching number.

Results for graphs with specified detour order.

Abstract

Let $K_{r_1,\ldots,r_s}$ denote the complete multipartite graph with class sizes $r_1,\ldots,r_s$ and let $K_s$ denote the complete graph of order $s$. In 2018, Luo determined the maximum number of $K_s$ in 2-connected graphs with a given circumference. Recently, Lu, Yuan and Zhang determined the maximum number of $K_{r_1,r_2}$ in 2-connected graphs with a given circumference, and Wang determined the maximum number of $K_s$ or $K_{r_1,1,\ldots,1}$ in graphs with given matching number. Motivated by these works, we determine the maximum number of $K_{r_1,\ldots,r_s}$ in $2$-connected graphs with given circumference and large minimum degree. The maximum number of $K_{r_1,\ldots,r_s}$ in graphs with given matching number and large minimum degree is also given. Consequently, we determine the maximum number of $K_{r_1,\ldots,r_s}$ in graphs with a given circumference or matching number. We also solve the corresponding problems for graphs with a given detour order.