On a conjecture on the order of cages with a given girth pair

Abstract

A (k; g, h)-graph is a k-regular graph of girth pair (g, h) where g is the girth of the graph, h is the length of a smallest cycle of different parity than g and g < h. A (k; g, h)-cage is a (k; g, h)-graph with the least possible number of vertices denoted n(k; g, h). Harary and Kovacs (1983) conjectured the inequality n(k; g, h) <= n(k, h) for all k >= 3, g >= 3, h >= g + 1. In this paper, we prove this conjecture for all (k; g, h)-cage with g odd provided that a bipartite (k, h)-cage exists. When g is even we prove the conjecture for h >= 2g - 1, provided that a bipartite (k, g)-cage exists. (C) 2015 Elsevier B.V. All rights reserved.