On 3-regular digraphs of girth 4

Abstract

We raise the conjecture that every 3-regular digraph, except the digraphs $D_7^3$, $D_8^3$ and $D_{2n}^2$ for every integer $n\ge 2$, which are defined in the introduction of the paper, contains two vertex disjoint directed cycles of different lengths and give support for this conjecture by proving that every 3-regular digraph of girth 4 contains two vertex disjoint directed cycles of different lengths.