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

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.