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arXiv · 1801.06169

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

Takayuki Hibi·Aki Mori·Hidefumi Ohsugi
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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.

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Takayuki Hibi, Aki Mori, Hidefumi Ohsugi. 2022-11-02. The number of $4$-cycles and the cyclomatic number of a finite simple graph. https://arxiv.org/abs/1801.06169

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