A case of the dijoin conjecture on inverting oriented graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915491943022592 |
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| author | Behague, Natalie Gaudart-Wifling, Patrick |
| author_facet | Behague, Natalie Gaudart-Wifling, Patrick |
| contents | For an oriented graph $D$, the inversion of $X\subseteq V(D)$ in $D$ is the graph obtained by reversing the orientation of all arcs with both ends in $X$. The inversion number $\mathrm{inv}(D)$ is the minimum number of inversions needed to obtain an acyclic oriented graph.
We show that the dijoin conjecture of Bang-Jensen, da Silva and Havet, that $\mathrm{inv}(D_1\rightarrow D_2)=\mathrm{inv}(D_1)+\mathrm{inv}(D_2)$, is true in the case where $\mathrm{inv}(D_1)=2$ and $\mathrm{inv}(D_2)$ is even. We also characterise the cases $\mathrm{inv}(D_1)=2$ and $\mathrm{inv}(D_2)$ odd, for which the conjecture does and does not hold. We then go on to show a similar result for n-joins, in doing so we prove a conjecture of Alon, Powierski, Savery, Scott and Wilmer. Our proofs build on the idea of tournament minimum rank, introduced by Behague, Johnston, Morrison and Ogden. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10232 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A case of the dijoin conjecture on inverting oriented graphs Behague, Natalie Gaudart-Wifling, Patrick Combinatorics 05C20, 05C35 For an oriented graph $D$, the inversion of $X\subseteq V(D)$ in $D$ is the graph obtained by reversing the orientation of all arcs with both ends in $X$. The inversion number $\mathrm{inv}(D)$ is the minimum number of inversions needed to obtain an acyclic oriented graph. We show that the dijoin conjecture of Bang-Jensen, da Silva and Havet, that $\mathrm{inv}(D_1\rightarrow D_2)=\mathrm{inv}(D_1)+\mathrm{inv}(D_2)$, is true in the case where $\mathrm{inv}(D_1)=2$ and $\mathrm{inv}(D_2)$ is even. We also characterise the cases $\mathrm{inv}(D_1)=2$ and $\mathrm{inv}(D_2)$ odd, for which the conjecture does and does not hold. We then go on to show a similar result for n-joins, in doing so we prove a conjecture of Alon, Powierski, Savery, Scott and Wilmer. Our proofs build on the idea of tournament minimum rank, introduced by Behague, Johnston, Morrison and Ogden. |
| title | A case of the dijoin conjecture on inverting oriented graphs |
| topic | Combinatorics 05C20, 05C35 |
| url | https://arxiv.org/abs/2509.10232 |