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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2605.24801 |
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| _version_ | 1866916042593271808 |
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| author | Gunasekara, Ajani De Vas |
| author_facet | Gunasekara, Ajani De Vas |
| contents | A transitive tournament is an acyclic orientation of a complete graph. We study decompositions and packings of the transitive tournament \(TT_n\) into connected two-arc motifs. The three motifs considered are chains, colliders, and forks, which are also fundamental local configurations in directed acyclic graphs. We first construct decompositions of \(TT_n\) into mixtures of these motifs whenever such decompositions exist. We then consider the corresponding pure packing problem for each individual motif. For \(H\) equal to a chain, a collider, or a fork, we determine the maximum number of arc-disjoint copies of \(H\) in \(TT_n\). These results give a precise extremal description of two-arc motif packings in transitive tournaments and suggest further questions on motif decompositions in broader classes of directed acyclic graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24801 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Chain--collider--fork Decompositions of Transitive Tournament Gunasekara, Ajani De Vas Combinatorics 05C20, 05C51, 05C20 A transitive tournament is an acyclic orientation of a complete graph. We study decompositions and packings of the transitive tournament \(TT_n\) into connected two-arc motifs. The three motifs considered are chains, colliders, and forks, which are also fundamental local configurations in directed acyclic graphs. We first construct decompositions of \(TT_n\) into mixtures of these motifs whenever such decompositions exist. We then consider the corresponding pure packing problem for each individual motif. For \(H\) equal to a chain, a collider, or a fork, we determine the maximum number of arc-disjoint copies of \(H\) in \(TT_n\). These results give a precise extremal description of two-arc motif packings in transitive tournaments and suggest further questions on motif decompositions in broader classes of directed acyclic graphs. |
| title | Chain--collider--fork Decompositions of Transitive Tournament |
| topic | Combinatorics 05C20, 05C51, 05C20 |
| url | https://arxiv.org/abs/2605.24801 |