On the order of graphs 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 < h. A (k; g, h)-cage is a (k; g, h)-graph with the least possible number of vertices denoted by n(k; g, h). In this paper we give a lower bound on n(k; g, h) and as a consequence we establish that every (k; 6)-cage is bipartite if it is free of odd cycles of length at most 2k - 1. This is a contribution to the conjecture claiming that every (k; g)-cage with even girth g is bipartite. We also obtain upper bounds on the order of (k; g, h)-graphs with g = 6, 8, 12. From the proofs of these upper bounds we obtain a construction of an infinite family of small (k; g, h)-graphs. In particular, the (3; 6, h)-graphs obtained for h = 7, 9, 11 are minimal. (C) 2013 Elsevier B.V. All rights reserved.