Tight constructions for reconfigurations of independent transversals
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917431178428416 |
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| author | Wdowinski, Ronen |
| author_facet | Wdowinski, Ronen |
| contents | For a graph $G$ and partition $\mathcal{U}$ of its vertex set, an independent transversal of $(G, \mathcal{U})$ is an independent set of $G$ that contains one vertex from each block of $\mathcal{U}$. Buys, Kang, and Ozeki studied when a reconfiguration graph on independent transversals of $(G,\mathcal{U})$ is connected, meaning any independent transversal can be transformed into any other one through a sequence of one-vertex modifications while always maintaining an independent transversal. Analogous to a theorem of Haxell, they proved that this is the case if $G$ has maximum degree $Δ$ and each block of $\mathcal{U}$ has size at least $2Δ$, except if the union of some $k \ge 1$ blocks of $\mathcal{U}$ induces $k$ disjoint copies of the complete bipartite graph $K_{Δ, Δ}$ in $G$. Solving one of their problems, we exactly characterize the partition structure in the latter exceptional instances of their theorem, showing that there is a rich variety of them but they are generated by a simple constructive procedure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21576 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tight constructions for reconfigurations of independent transversals Wdowinski, Ronen Combinatorics For a graph $G$ and partition $\mathcal{U}$ of its vertex set, an independent transversal of $(G, \mathcal{U})$ is an independent set of $G$ that contains one vertex from each block of $\mathcal{U}$. Buys, Kang, and Ozeki studied when a reconfiguration graph on independent transversals of $(G,\mathcal{U})$ is connected, meaning any independent transversal can be transformed into any other one through a sequence of one-vertex modifications while always maintaining an independent transversal. Analogous to a theorem of Haxell, they proved that this is the case if $G$ has maximum degree $Δ$ and each block of $\mathcal{U}$ has size at least $2Δ$, except if the union of some $k \ge 1$ blocks of $\mathcal{U}$ induces $k$ disjoint copies of the complete bipartite graph $K_{Δ, Δ}$ in $G$. Solving one of their problems, we exactly characterize the partition structure in the latter exceptional instances of their theorem, showing that there is a rich variety of them but they are generated by a simple constructive procedure. |
| title | Tight constructions for reconfigurations of independent transversals |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.21576 |