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

Ryan C. Bunge1, Saad I. El-Zanati1, Jessica Klister2, Dan Roberts3, Catherine Ruddell3
1Illinois State University Normal, Illinois, U.S.A.
2University of Wisconsin-La Crosse La Crosse, Wisconsin, U.S.A.
3Illinois Wesleyan University Eastern Illinois University Bloomington, [Hinois, U.S.A. Charleston, Illinois, U.S.A.

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 \( K_{nx+1}^* \) for every positive integer \( x \). We also find several classes of bipartite digraphs that admit an ordered directed \(\rho\)-labeling.