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 (2011) 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.