Bounds on the order of biregular graphs with even girth at least 8

Let 2 ⩽ r < m and g ⩾ 4 even be three positive integers. A graph with a degree set { r , m } , girth g and minimum order is called a bi-regular cage or an ( { r , m } ; g ) -cage, and its order is denoted by n ( { r , m } ; g ) . In this paper we obtain constructive upper bounds on n ( { r , m } ; g ) for some values of r , m and even girth at least 8.