Tournament completions of bipartite tournaments and their augmented directed cycles

TL;DR

This paper studies how to complete bipartite tournaments into larger tournaments with specific properties regarding augmented directed cycles, focusing on the existence of unique augmented 3-cycles and absence of augmented 4-cycles.

Contribution

It introduces a new variant of the orientation completion problem, characterizing bipartite tournaments that admit such specialized tournament completions.

Findings

Identifies conditions for bipartite tournaments to have a completion with one augmented 3-dicycle.

Establishes the absence of augmented 4-dicycles in certain tournament completions.

Provides a framework for understanding cycle augmentation in tournament completions.

Abstract

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 $A(C)\not\subseteq A(D)$, 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..