New families of small regular graphs of girth 5

TL;DR

This paper introduces new small regular graphs of girth 5, improving known upper bounds for the Cage Problem by applying existing construction techniques.

Contribution

It constructs new regular graphs of girth 5 for degrees where cages are previously unknown, advancing the understanding of the Cage Problem.

Findings

New small regular graphs of girth 5 constructed

Improved upper bounds for the Cage Problem

Enhanced understanding of graphs with girth 5

Abstract

In this paper we are interested in the {\it{Cage Problem}} that consists in constructing regular graphs of given girth $g$ and minimum order. We focus on girth $g=5$, where cages are known only for degrees $k \le 7$. We construct regular graphs of girth $5$ using techniques exposed by Funk [Note di Matematica. 29 suppl.1, (2009) 91 - 114] and Abreu et al. [Discrete Math. 312 (2012), 2832 - 2842] to obtain the best upper bounds known hitherto. The tables given in the introduction show the improvements obtained with our results.