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

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 = 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.Peer Reviewe