Optimal packings and coverings of \(\lambda DK_\nu\) with \(k\)-circuits

Let \(\lambda DK_v\). denote the complete directed multigraph with \(v\). vertices, where any two distinct vertices \(x\). and \(y\). are joined by \(\lambda\). arcs \((x,y)\). and \(\lambda\). arcs \((y,x)\).. By a \(k\).-circuit we mean a directed cycle of length \(k\).. In this paper, we consider the problem of constructing maximal packings and minimal coverings of \(\lambda DK_v\). with \(k\).-circuits. Using the leave-arcs graph of packing and the repeat-arcs graph of covering, we give a unified method for finding packings and coverings. Also, we completely solve the existence of optimal packings and coverings for \(5 \leq k \leq 14\). and any \(\lambda\).