On generalized cages

Abstract

In this paper, we give a generalization to the well‐known cage concept as follows: We define an (r,g,k)‐cage to be an r(k − 1)‐regular graph of minimum order in which every clique has k vertices, every vertex is in r cliques, and the minimum length of a cycle with edges from distinct cliques is g. Clearly, when g ≥ 4 and k = 2, an (r,g, 2)‐cage is just a usual (r,g)‐cage. Here we prove that an (r,g,k)‐cage always exists for g ≥ 4. We examine the lower bounds for the order of (r,g,k)‐cages and also explore some connections of these generalized cages with generalized polygons, strongly regular graphs, and Moore geometries, which give rise to certain (r,g,k)‐cages. © 1996 John Wiley & Sons, Inc.