Coverings of the complete directed graph with \(k\)-circuits

A covering of the complete directed symmetric graph \(DK_v\) by \(m\)-circuits, denoted by \((v,m) – {DCC}\), is a family of \(m\)-circuits in \(DK_v\) whose union is \(DK_v\). 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 \(v \geq m\). In this paper, the problem is reduced to the case \(m+5 \leq v \leq 2m – 4\), for any fixed even integer \(m \geq 4\).

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.