On the Monotonicity of (k;g,h)-graphs

Abstract

A (k;g)-graph is a k-regular graph with girth g. A (k;g)-cage is a (k;g)-graph with the least possible number of vertices. Let f (k;g) denote the number of vertices in a (k;g)-cage. The girth pair of a graph gives the length of a shortest odd and a shortest even cycle. A k-regular graph with girth pair (g,h) is called a (k;g,h)-graph. A (k;g,h)-cage is a (k;g,h)-graph with the least possible number of vertices. Let f(k;g,h) denote the number of vertices in a (k;g,h)-cage. In this paper, we prove the following strict inequality f (k;h–1,h)<f(k,h), k≥3, h≥4.