The number of $4$-cycles and the cyclomatic number of a finite simple graph

TL;DR

This paper establishes upper bounds on the number of 4-cycles in finite simple graphs, differentiating between bipartite and non-bipartite cases, and offers combinatorial proofs for these bounds.

Contribution

It provides new upper bounds on 4-cycle counts in graphs and presents concise combinatorial proofs for both bipartite and non-bipartite graphs.

Findings

Non-bipartite graphs have at most inom{m-n+1}{2} 4-cycles.

Bipartite graphs have at most inom{m-n+2}{2} 4-cycles.

The bounds are proven using combinatorial methods.

Abstract

Let $G$ be a finite connected simple graph with $n$ vertices and $m$ edges. We show that, when $G$ is not bipartite, the number of $4$-cycles contained in $G$ is at most $\binom{m-n+1}{2}$. We further provide a short combinatorial proof of the bound $\binom{m-n+2}{2}$ which holds for bipartite graphs.