On ordered directed $\rho$-labelings of bipartite digraphs and cyclic digraph decompositions

Abstract

It is known that an ordered $\rho$-labeling of a bipartite graph $G$ with $n$ edges yields a cyclic $G$-decomposition of $K_{2nx+1}$ for every positive integer $x$. We extend the concept of an ordered $\rho$-labeling to bipartite digraphs and show that an ordered directed $\rho$-labeling of a bipartite digraph $D$ with $n$ arcs yields a cyclic $D$-decomposition of $K_{nx+1}^{\ast }$ for every positive integer $x$. We also find several classes of bipartite digraphs that admit an ordered directed $\rho$-labeling.