Mixed Cages

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$g\geq 3$,andforany$z\geq 1$,$r=1$and$g=4$.