A simple proof for the lower bound of the girth of graphs $D(n, q)$

TL;DR

This paper presents a new, simplified proof establishing the exact girth of the graphs $D(n, q)$, which is crucial for understanding their cycle structure and extremal properties.

Contribution

It provides a concise and accessible proof for the lower bound of the girth of $D(n, q)$ graphs, improving upon previous more complex demonstrations.

Findings

Girth of $D(n, q)$ is $n + 5$ for odd $n$

Girth of $D(n, q)$ is $n + 4$ for even $n$

Proof simplifies understanding of cycle lengths in these graphs

Abstract

The components of the graphs $D(n, q)$ provide the best-known general lower bound for the number of edges in a graph with $n$ vertices and no cycles of length less than $g$. In this paper, we give a new, short, and simpler proof of the fact that the length of the shortest cycle appearing in $D(n, q)$ is $n + 5$ when $n$ is odd, and $n + 4$ when $n$ is even.