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

TL;DR

This paper introduces a new algorithm based on hill-climbing heuristics to improve lower bounds on the maximum size of graphs with girth at least 5 for a range of vertex counts, surpassing previous bounds.

Contribution

The paper develops a novel heuristic algorithm that propagates good graph patterns across different sizes to improve lower bounds on graph size with girth 5.

Findings

Improved lower bounds for most values of n between 74 and 198.

Effective propagation of graph patterns across different n values.

Achieved better bounds than existing methods for the majority of tested n.

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 $n$ ($n=96,97$).