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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2022
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2205.14904 |
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- Let $G$ be a $q$-regular bipartite graph with bipartition $(U,V)$. It was proved by Lu, Wang, and Yan in 2020 that $G$ has a spanning subgraph $H$ such that each vertex of $U$ has degree 1 in $H$, and each vertex of $V$ has degree distinct from 1 in $H$. We extend the result to multigraphs, under the condition that $q$ is a prime power and the number of perfect matchings of $G$ is not divisible by $q$. The condition on the number of perfect matchings is necessary for multigraphs. We conclude with a conjecture on the limiting distribution of the number of perfect matchings modulo $q$ in a random bipartite $q$-regular graph.