On cycle covers of infinite bipartite graphs
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915226290487296 |
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| author | Aurichi, Leandro Júnior, Paulo Magalhães Zdomskyy, Lyubomyr |
| author_facet | Aurichi, Leandro Júnior, Paulo Magalhães Zdomskyy, Lyubomyr |
| contents | Given a graph $G$ and a subset $X$ of vertices of $G$ with size at least two, we denote by $N^2_G(X)$ the set of vertices of $G$ that have at least two neighbors in $X$. We say that a bipartite graph $G$ with sides $A$ and $B$ satisfies the double Hall property if for every subset $X$ of vertices of $A$ with size at least 2, $\vert N^2_G(X)\vert \geq \vert X\vert$. Salia conjectured that if $G$ is a bipartite graph that satisfies the double Hall property, then there exists a cycle in $G$ that covers all vertices of $A$. In this work, we study this conjecture restricted to infinite graphs. For this, we use the definition of ends and infinite cycles. It is simple to see that Salia's conjecture is false for infinite graphs in general. Consequently, all our results are partial. Under certain hypothesis it is possible to obtain a collection of pairwise disjoint 2-regular subgraphs that covers $A$. We show that if side $B$ is locally finite and side $A$ is countable, then the conjecture is true. Furthermore, assuming the conjecture holds for finite graphs, we show that it holds for infinite graphs with a restriction on the degree of the vertices of $B$. This result is inspired by the result obtained by Barát, Grzesik, Jung, Nagy and Pálvölgyi for finite graphs. Finally, we also show that if Salia's conjecture holds for some cases of infinite graphs, then the conjecture about finite graphs presented by Lavrov and Vandenbussche is true. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02816 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On cycle covers of infinite bipartite graphs Aurichi, Leandro Júnior, Paulo Magalhães Zdomskyy, Lyubomyr Combinatorics General Topology 05C05 05C38 05C63 54H99 Given a graph $G$ and a subset $X$ of vertices of $G$ with size at least two, we denote by $N^2_G(X)$ the set of vertices of $G$ that have at least two neighbors in $X$. We say that a bipartite graph $G$ with sides $A$ and $B$ satisfies the double Hall property if for every subset $X$ of vertices of $A$ with size at least 2, $\vert N^2_G(X)\vert \geq \vert X\vert$. Salia conjectured that if $G$ is a bipartite graph that satisfies the double Hall property, then there exists a cycle in $G$ that covers all vertices of $A$. In this work, we study this conjecture restricted to infinite graphs. For this, we use the definition of ends and infinite cycles. It is simple to see that Salia's conjecture is false for infinite graphs in general. Consequently, all our results are partial. Under certain hypothesis it is possible to obtain a collection of pairwise disjoint 2-regular subgraphs that covers $A$. We show that if side $B$ is locally finite and side $A$ is countable, then the conjecture is true. Furthermore, assuming the conjecture holds for finite graphs, we show that it holds for infinite graphs with a restriction on the degree of the vertices of $B$. This result is inspired by the result obtained by Barát, Grzesik, Jung, Nagy and Pálvölgyi for finite graphs. Finally, we also show that if Salia's conjecture holds for some cases of infinite graphs, then the conjecture about finite graphs presented by Lavrov and Vandenbussche is true. |
| title | On cycle covers of infinite bipartite graphs |
| topic | Combinatorics General Topology 05C05 05C38 05C63 54H99 |
| url | https://arxiv.org/abs/2504.02816 |