Biregular Cages of Girth Five

Abstract

Let $2 \le r < m$ 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 [M. Abreu,  G. Araujo-Pardo, C. Balbuena, D. Labbate. Families of Small Regular Graphs of Girth $5$. Discrete Math. 312 (2012) 2832--2842] 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.