Coverings of the complete directed graph with $k$-circuits

Abstract

A covering of the complete directed symmetric graph $D{K}_v$ by $m$-circuits, denoted by $(v,m)–DCC$, is a family of $m$-circuits in $D{K}_v$ whose union is $D{K}_v$. The covering number $C(v,m)$ is the minimum number of $m$-circuits in such a covering.The covering problem is to determine the value $C(v,m)$ for every integer $v\ge m$. In this paper, the problem is reduced to the case $m+5\le v\le 2m–4$, for any fixed even integer $m\ge 4$.In particular, the values of $C(v,m)$ are completely determined for $m=12,14,$ and $16$. Additionally, a directed construction of optimal $(6k+11,4k+6)–DCC$ is given.