Biregular cages of girth five

TL;DR

This paper introduces new infinite families of biregular cages with girth five by extending existing graph operations, and constructs specific semiregular cages with small orders.

Contribution

It generalizes reduction and amalgam operations to produce two new infinite families of biregular cages and two semiregular cages, expanding known cage constructions.

Findings

Constructed $({r,2r-3};5)$-cages for all $r=q+1$ with $q$ prime power.

Constructed $({r,2r-5};5)$-cages for all $r=q+1$ with $q$ prime.

Built semiregular cages with 31 and 43 vertices for $r=5$ and $r=6$.

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.