Walecki tournaments with an arc that lies in a unique directed triangle
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911945362243584 |
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| author | Morris, Joy |
| author_facet | Morris, Joy |
| contents | A Walecki tournament is any tournament that can be formed by choosing an orientation for each of the Hamilton cycles in the Walecki decomposition of a complete graph on an odd number of vertices. In this paper, we show that if some arc in a Walecki tournament on at least $7$ vertices lies in exactly one directed triangle, then there is a vertex of the tournament (the vertex typically labelled $*$ in the decomposition) that is fixed under every automorphism of the tournament. Furthermore, any isomorphism between such Walecki tournaments maps the vertex labelled $*$ in one to the vertex labelled $*$ in the other.
We also show that among Walecki tournaments with a signature of even length $2k$, of the $2^{2k}$ possible signatures, at least $2^k$ produce tournaments that have an arc that lies in a unique directed triangle (and therefore to which our result applies). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03807 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Walecki tournaments with an arc that lies in a unique directed triangle Morris, Joy Combinatorics 05C20, 05C38, 05C45, 05C60 A Walecki tournament is any tournament that can be formed by choosing an orientation for each of the Hamilton cycles in the Walecki decomposition of a complete graph on an odd number of vertices. In this paper, we show that if some arc in a Walecki tournament on at least $7$ vertices lies in exactly one directed triangle, then there is a vertex of the tournament (the vertex typically labelled $*$ in the decomposition) that is fixed under every automorphism of the tournament. Furthermore, any isomorphism between such Walecki tournaments maps the vertex labelled $*$ in one to the vertex labelled $*$ in the other. We also show that among Walecki tournaments with a signature of even length $2k$, of the $2^{2k}$ possible signatures, at least $2^k$ produce tournaments that have an arc that lies in a unique directed triangle (and therefore to which our result applies). |
| title | Walecki tournaments with an arc that lies in a unique directed triangle |
| topic | Combinatorics 05C20, 05C38, 05C45, 05C60 |
| url | https://arxiv.org/abs/2407.03807 |