Counting Cycles in Graphs with Bounded Circumference

Abstract

For an integer $L\ge 2$, let $a=\lfloor L/2\rfloor$. Let $H(n,L)$ be the join of $K_a$ and an independent set of order $n-a$, with one extra edge in the independent set when $L$ is odd. We prove that, for fixed integers $q\ge 4$ and $L>q$, and for all sufficiently large $n$, the graph $H(n,L)$ maximizes the number of copies of $C_q$ among all $n$-vertex graphs of circumference at most $L$. This settles a conjecture of Zhu, Győri, He, Lv, Salia and Xiao~[Bull. Lond. Math. Soc. 55 (2023)]. For even $q\ge 6$, we also prove the boundary case $L=q$. We further determine the corresponding maximum when a long path is forbidden.