Optimal Packings and Coverings of $\lambda D{K}_{\nu }$ with $k$-circuits

Abstract

Let $\lambda D{K}_v$. 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 $\lambda D{K}_v$. 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 $5\le k\le 14$. and any $\lambda$.