Constructions for Cubic Graphs with Large Girth

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 μ0(g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value μ0(g) is attained by a graph whose girth is exactly g. The values of μ0(g) when 3 ≤ g ≤ 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 ≥ 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 {Gi} for which the number of vertices ni and the girth gi are such that ni ≤ 2i 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. MR Subject Numbers: 05C25, 05C35, 05C38.