On the Spectra of Bipartite Directed Subgraphs of $K_n^{\ast }$

Abstract

The complete directed graph of order $n$, denoted $K_n^{\ast }$, is the directed graph on $n$ vertices that contains the arcs $(u,v)$ and $(v,u)$ for every pair of distinct vertices $u$ and $v$. For a given directed graph $D$, the set of all $n$ for which $K_n^{\ast }$ admits a $D$-decomposition is called the spectrum of $D$. In this paper, we find the spectrum for each bipartite subgraph of $K_4^{\ast }$ with 5 or fewer arcs.