Tournament completions of bipartite tournaments and their augmented directed cycles

A tournament $T$ is a tournament completion of a bipartite tournament $D$ if $D$ is a spanning subdigraph of $T$, i.e., $V(D)=V(T)$ and $A(D)\subseteq A(T)$. If $C$ is a $k$-dicycle (i.e., directed cycle of length $k$) in a tournament completion $T$ of $D$ and $C$ is not a dicycle in $D$, i.e., $A(C)\subseteq A(T)$ and $A(C)\subseteq ̸A(D)$, then we call $C$ an augmented $k$-dicycle of $T$. In this paper, we investigate the families of bipartite tournaments for which there exists a tournament completion with exactly one augmented $3$-dicycle and with no augmented $4$-dicycles. Our investigation may be viewed as a variant of the orientation completion problem initiated by Bang-Jensen et al…