Constructions of biregular cages of girth five

Let 2 ⩽ r < m and g be positive integers. An ( { r , m } ; g ) –graph (or biregular graph) is a graph with degree set { r , m } and girth g, and an ( { r , m } ; g ) –cage (or biregular cage) is an ( { r , m } ; g ) –graph of minimum order n ( { r , m } ; g ) . If m = r + 1 , an ( { r , m } ; g ) –cage is said to be a semiregular cage. s extended abstract we construct two infinite families of biregular cages and two semiregular cages, obtained from the incidence graphs of an affine and a biaffine plane by generalizations of the reduction and graph amalgam operations from [M. Abreu, G. Araujo-Pardo, C. Balbuena, D. Labbate. Families of Small Regular Graphs of Girth 5. Discrete Math. 312(18) (2012) 2832–2842].