Mixed Cages: monotony, connectivity and upper bounds

TL;DR

This paper investigates the structural properties and bounds of mixed cages, specifically focusing on their monotonicity, connectivity, and minimal order for given parameters, and presents new constructions including a specific example of a mixed cage.

Contribution

It establishes the monotonicity of the order function for certain degrees, proves connectivity properties, and provides new bounds and explicit constructions of mixed cages.

Findings

The order of mixed cages is monotonic with respect to girth for certain degrees.

Mixed cages are proven to be 2-connected and strongly connected.

Explicit construction of a [10,3;5]-mixed cage with 50 vertices.

Abstract

A \emph{$[z, r; g]$-mixed cage} is a mixed graph $z$-regular by arcs, $r$-regular by edges, with girth $g$ and minimum order. %In this paper we study structural properties of mixed cages: Let $n[z,r;g]$ denote the order of a $[z,r;g]$-mixed cage. In this paper we prove that $n[z,r;g]$ is a monotonicity function, with respect of $g$, for $z\in \{1,2\}$, and we use it to prove that the underlying graph of a $[z,r;g]$-mixed cage is 2-connected, for $z\in \{1,2\}$. We also prove that $[z,r;g]$-mixed cages are strong connected. We present bounds of $n[z,r;g]$ and constructions of $[z,r;5]$-mixed graphs and show a $[10,3;5]$-mixed cage of order $50$.