Improved lower bounds on the maximum size of graphs with girth 5
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912525941997568 |
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| author | Goedgebeur, Jan Jooken, Jorik Joret, Gwenaël Eede, Tibo Van den |
| author_facet | Goedgebeur, Jan Jooken, Jorik Joret, Gwenaël Eede, Tibo Van den |
| contents | 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 $n$ ($n=96,97$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05562 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved lower bounds on the maximum size of graphs with girth 5 Goedgebeur, Jan Jooken, Jorik Joret, Gwenaël Eede, Tibo Van den Combinatorics Discrete Mathematics 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 $n$ ($n=96,97$). |
| title | Improved lower bounds on the maximum size of graphs with girth 5 |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2508.05562 |