Tournament completions of bipartite tournaments and their augmented directed cycles
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910481705336832 |
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| author | Wong, H. W. Willie |
| author_facet | Wong, H. W. Willie |
| contents | 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.. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18453 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tournament completions of bipartite tournaments and their augmented directed cycles Wong, H. W. Willie Combinatorics 05C20, 05C38 G.2.2 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.. |
| title | Tournament completions of bipartite tournaments and their augmented directed cycles |
| topic | Combinatorics 05C20, 05C38 G.2.2 |
| url | https://arxiv.org/abs/2405.18453 |