New upper bounds on the order of mixed cages of girth 6

TL;DR

This paper introduces new upper bounds for the minimum order of mixed graphs with girth 6, improving previous bounds by constructing specific infinite families based on prime power parameters.

Contribution

The authors construct new infinite families of mixed graphs with girth 6 that establish improved upper bounds on their minimum order, extending prior results.

Findings

Constructed families of mixed graphs with girth 6 for even prime powers.

Established upper bounds of 4q^2-4 for these graphs.

Provided bounds depending on the parity of (q-3)/2 for odd prime powers.

Abstract

A $[z,r;g]$-mixed cage is a mixed graph of minimum order such that each vertex has $z$ in-arcs, $z$ out-arcs, $r$ edges, and it has girth $g$, and the minimum order for $[z,r;g]$-mixed graphs is denoted by $n[z,r;g]$. In this paper, we present an infinite family of mixed graphs with girth $6$, that improves, in some cases, the families that we give in G. Araujo-Pardo and L. Mendoza-Cadena. \textit{On Mixed Cages of girth 6}, arXiv:2401.14768v2. In particular, if $q$ is an even prime power we construct a family of graphs that satisfies $n[\frac{q}{4},q;6]\leq 4q^2-4$, and if $q$ is an odd prime power, and $\frac{q-3}{2}$ is odd then our family satisfies that $n[\frac{q-1}{4},q;6]\leq 4q^2-4$, otherwise $n[\frac{q-3}{4},q;6]\leq 4q^2-4$.