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 Kóvacs (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.