Distances in graphs of girth 6 and generalised cages

Abstract

In this paper we present bounds on the radius and diameter of graphs of girth at least 6 and for (C4,C5)-free graphs, i.e., graphs not containing cycles of length 4 or 5. We show that the diameter of a graph G of girth at least 6 is at most 3nδ2−δ+1−1, and the radius is at most 3n2(δ2−δ+1)+10, where n is the order and δ the minimum degree of G. If δ−1 is a prime power, then both bounds are sharp apart from an additive constant.For graphs of large maximum degree Δ, we show that these bounds can be improved to 3n−Δδδ2−δ+1−3(δ−1)Δ(δ−2)δ2−δ+1+10 for the diameter, and 3n−3Δδ2(δ2−δ+1)−3(δ−1)Δ(δ−2)2(δ2−δ+1)+22 for the radius. We further show that only slightly weaker bounds hold for (C4,C5)-free graphs.As a by-product we obtain a result on a generalisation of cages. For given δ,Δ∈N with Δ≥δ let n(δ,Δ,g) be the minimum order of a graph of girth g, minimum degree δ and maximum degree Δ. Then n(δ,Δ,6)≥Δδ+(δ−1)Δ(δ−2)+32. If δ−1 is a prime power, then there exist infinitely many values of Δ such that, for δ constant and Δ large, n(δ,Δ,6)=δΔ+O(Δ).