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arXiv · 2508.05562

Improved lower bounds on the maximum size of graphs with girth 5

Jan Goedgebeur·Jorik Jooken·Gwenaël Joret·Tibo Van den Eede
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Abstract

We present a new algorithm for improving lower bounds on $ex(n;\{C_3,C_4\})$, the maximum size (number of edges) of an $n$-vertex graph of girth at least 5. The core of our algorithm is a variant of a hill-climbing heuristic introduced by Exoo, McKay, Myrvold and Nadon (2011) to find small cages. Our algorithm considers a range of values of $n$ in multiple passes. In each pass, the hill-climbing heuristic for a specific value of $n$ is initialized with a few graphs obtained by modifying near-extremal graphs previously found for neighboring values of $n$, allowing to `propagate' good patterns that were found. Focusing on the range $n\in \{74,75, \dots, 198\}$, which is currently beyond the scope of exact methods, our approach yields improvements on existing lower bounds for $ex(n;\{C_3,C_4\})$ for all $n$ in the range, except for two values of $n$ ($n=96,97$).

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Jan Goedgebeur, Jorik Jooken, Gwenaël Joret, Tibo Van den Eede. 2025-08-07. Improved lower bounds on the maximum size of graphs with girth 5. https://arxiv.org/abs/2508.05562

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