# Mixed Cages

## TL;DR

This paper introduces and studies $[z, r; g]$-mixed cages, a new class of graphs with specific regularity and girth properties, providing existence proofs, constructions, and bounds for various parameters.

## Contribution

It defines the concept of $[z, r; g]$-mixed cages, proves their existence, and presents the first results for certain parameter sets, including explicit families and bounds.

## Key findings

- Existence of $[z, r; g]$-mixed cages established.
- Constructed families of mixed cages for specific parameters.
- Derived bounds for the order of mixed cages.

## Abstract

We introduce the notion of a $[z, r; g]$-mixed cage. A $[z, r; g]$-mixed cage is a mixed graph $G$, $z$-regular by arcs, $r$-regular by edges, with girth $g$ and minimum order. In this paper we prove the existence of $[z, r ;g]$-mixed cages and exhibit families of mixed cages for some specific values. We also give lower and upper bounds for some choices of $z, r$ and $g$. In particular we present the first results on $[z,r;g]$- mixed cages for $z=1$ and any $r\geq 1$ and $g\geq 3$, and for any $z\geq 1$, $r=1$ and $g=4$.