Extremal edge-girth-regular graphs

Abstract

An edge-girth-regular egr(v, k, g, lambda)-graph Gamma is a k-regular graph of order v and girth g in which every edge is contained in lambda distinct g-cycles. Edge-girth-regularity is shared by several interesting classes of graphs which include edge- and arctransitive graphs, Moore graphs, as well as many of the extremal k-regular graphs of prescribed girth or diameter. Infinitely many egr(v, k, g, lambda)-graphs are known to exist for sufficiently large parameters (k, g, lambda), and in line with the well-known Cage Problem we attempt to determine the smallest graphs among all edge-girth-regular graphs for given parameters (k, g, lambda). To facilitate the search for egro(v, k, g, lambda)-graphs of the smallest possible orders, we derive lower bounds in terms of the parameters k, g and lambda. We also determine the orders of the smallest egro(v, k, g, lambda)-graphs for some specific parameters (k, g, lambda), and address the problem of the smallest possible orders of bipartite edge-girth-regular graphs.