A Note on Mixed Cages of Girth 5
Gabriela Araujo-Pardo, Lydia Mirabel Mendoza-Cadena
TL;DR
This work addresses constructing small-order mixed graphs with girth $5$ (mixed cages) and improves the upper bounds on $n[z,q;5]$ for suitable $q$. It leverages the incidence graph framework of elliptic semiplanes of type $L$, building on $G_q$, derived from $PG(2;q)$, and employs primitive elements and circulant digraphs to augment the structure. For prime-power $q\ge 7$ with $q-1=4k+R$ ($R\in\{1,\dots,5\}$), it explicitly constructs a $[k,q;5]$-mixed graph of order $2q^2-2$, establishing $n[k,q;5]\le 2q^2-2$ and improving previous bounds known for prime $q$. The paper also situates these results within a broader lower-bound framework, $q^2+q+4z+1 \le n[z,q;5] \le 2q^2-2$, highlighting the value of elliptic-semiplane-based approaches for mixed cages and extending the scope to prime-power $q$.
Abstract
A mixed regular graph is a graph where every vertex has $z$ incoming arcs, $z$ outgoing arcs, and $r$ edges; furthermore, if it has girth $g$, we say that the graph is a \emph{$[z,r;g]$-mixed graph}. A \emph{$[z,r;g]$-mixed cage} is a $[z,r;g]$-mixed graph with the smallest possible order. In this note, we give a family of $[z,q;5]$-mixed graphs for $q\geq 7$ power of prime and $q-1\leq 4z+R$ with $z\geq 1$ and $R \in \{1,\ldots,5\}$. This provides better upper bounds on the order of mixed cages until this moment.