The diameter of dense random regular graphs

Abstract

There is a tight upper bound on the order (the number of vertices) of any d-regular graph of diameter D, known as the Moore bound in graph theory. This bound implies a lower bound D0(n; d) on the diameter of any d-regular graph of order n. Actually, the diameter diam(Gn;d) of a random d-regular graph Gn;d of order n is known to be asymptotically "optimal" as n → ∞. Bollobás and de la Vega (1982) proved that diam(Gn;d) = (1 + o(1))D0(n; d) = (1 + o(1)) logd-1 n holds w.h.p. (with high probability) for fixed d 3, whereas there exists a gap diam(Gn;d) D0(n; d) = O( log log n). In this paper, we investigate the gap diam(Gn;d) D0(n; d) for d = (ϵ+o(1)) n where 2 (0; 1) and > 0 are arbitrary constants. We prove that diam(Gn;d) = 1+1 holds w.h.p. for such d. Our result implies that the gap is 1 if1 is an integer and d- n, and is 0 otherwise. One can easily obtain that diam(Gn;d) ≥ -1 + 1 holds w.h.p. by using the embedding theorem due to Dudek et al. (2017). Our critical contribution is to show that diam(Gn;d)≥ [α-1] + 1 holds w.h.p. by the analysis of the distances of fixed vertex pairs.