On balanced biregular cages

TL;DR

This paper explores the concept of balanced biregular cages, introducing the problem, constructing graphs from finite geometries, and establishing bounds for their parameters.

Contribution

It defines balanced biregular cages, provides bounds, and constructs small examples using incidence graphs from finite geometries.

Findings

Constructed small balanced biregular graphs from incidence graphs.

Established lower and upper bounds for degree and girth.

Identified some graphs as balanced biregular cages.

Abstract

In this paper, we introduce a problem closely related to the {\emph{Cage Problem}}. We are interested in {\emph{Balanced Biregular Cages}}, which are the smallest biregular graphs of fixed girth that have the same number of vertices of one degree as the other. We introduce the graphs and obtain lower and upper bounds for some values of degree and girth. In particular, we construct relatively small balanced biregular graphs from incidence graphs of finite projective, affine, and biaffine planes and we show that some of the obtained graphs are balanced biregular cages.