Monotonicity of the order of -cages

A ( D ; g ) -cage is a graph having degree set D , girth g , and the minimum possible number of vertices, which is denoted by n ( D ; g ) . When D = { r } the corresponding ( { r } ; g ) -cage is clearly r -regular, and is called an ( r ; g ) -cage. In this work we prove that if g < g ′ then n ( D ; g ) < n ( D ; g ′ ) under certain requirements on the elements of the degree set D or on the girth g .