Constructions for Cubic Graphs with Large Girth

Abstract

The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a well-defined integer $\mu_0(g)$, the smallest number of vertices for which a cubic graph with girth at least $g$ exists, and furthermore, the minimum value $\mu_0(g)$ is attained by a graph whose girth is exactly $g$. The values of $\mu_0(g)$ when $3 \le g \le 8$ have been known for over thirty years. For these values of $g$ each minimal graph is unique and, apart from the case $g=7$, a simple lower bound is attained. This paper is mainly concerned with what happens when $g \ge 9$, where the situation is quite different. Here it is known that the simple lower bound is attained if and only if $g=12$. A number of techniques are described, with emphasis on the construction of families of graphs $\{ G_i\}$ for which the number of vertices $n_i$ and the girth $g_i$ are such that $n_i\le 2^{cg_i}$ for some finite constant $c$. The optimum value of $c$ is known to lie between $0.5$ and $0.75$. At the end of the paper there is a selection of open questions, several of them containing suggestions which might lead to improvements in the known results. There are also some historical notes on the current-best graphs for girth up to 36.