On the number of cycles in a graph with restricted cycle lengths

TL;DR

This paper investigates the maximum number of cycles in graphs with restricted cycle lengths, providing asymptotic bounds and exact values for specific cases in both undirected and directed graphs.

Contribution

It establishes asymptotic formulas for the maximum number of cycles in L-cycle graphs and characterizes extremal graphs for single-element cycle length sets.

Findings

In undirected graphs, the maximum number of cycles scales as n^{floor(k/ell)} for fixed L.

In directed graphs, the maximum number of cycles is approximately ((n-1)/(k-1))^{k-1}.

Exact maximum cycle counts are determined for graphs with a single cycle length.

Abstract

Let $L$ be a set of positive integers. We call a (directed) graph $G$ an $L$\emph{-cycle graph} if all cycle lengths in $G$ belong to $L$. Let $c(L,n)$ be the maximum number of cycles possible in an $n$-vertex $L$-cycle graph (we use $\vec{c}(L,n)$ for the number of cycles in directed graphs). In the undirected case we show that for any fixed set $L$, we have $c(L,n)=\Theta_L(n^{\lfloor k/\ell \rfloor})$ where $k$ is the largest element of $L$ and $2\ell$ is the smallest even element of $L$ (if $L$ contains only odd elements, then $c(L,n)=\Theta_L(n)$ holds.) We also give a characterization of $L$-cycle graphs when $L$ is a single element. In the directed case we prove that for any fixed set $L$ we have $\vec{c}(L,n)=(1+o(1))(\frac{n-1}{k-1})^{k-1}$, where $k$ is the largest element of $L$. We determine the exact value of $\vec{c}(\{k\},n)$ for every $k$ and characterize all graphs attaining this maximum.