On the order of graphs 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 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.