# Biregular Cages of Girth Five

Keywords:
biregular, cage, girth

### 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.