On the Spectra of Bipartite Directed Subgraphs of \({K}_n^*\)

The complete directed graph of order \(n\), denoted \({K}_n^*\), 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 \({K}_n^*\) admits a \(D\)-decomposition is called the spectrum of \(D\). In this paper, we find the spectrum for each bipartite subgraph of \({K}_4^*\) with 5 or fewer arcs.