Biregular cages of girth five (Q1953464)

Summary: Let \(2 \leq r < m\) 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. In this paper we generalize the reduction and graph amalgam operations from [\textit{M. Abreu} et al., Discrete Math. 312, No. 18, 2832--2842 (2012; Zbl 1248.05169)] on the incidence graphs of an affine and a biaffine plane obtaining two new infinite families of biregular cages and two new semiregular cages. The constructed new families are \((\{r,2r-3\};5)\)-cages for all \(r=q+1\) with \(q\) a prime power, and \((\{r,2r-5\};5)\)-cages for all \(r=q+1\) with \(q\) a prime. The new semiregular cages are constructed for \(r=5\) and 6 with \(31\) and \(43\) vertices respectively.