Paired $(n-1)$-to-$(n-1)$ disjoint path covers in bipartite transposition-like graphs
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912545821949952 |
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| author | Coleman, Anna Fischberg, Gabrielle Gong, Charles Harrington, Joshua Wong, Tony W. H. |
| author_facet | Coleman, Anna Fischberg, Gabrielle Gong, Charles Harrington, Joshua Wong, Tony W. H. |
| contents | A paired $k$-to-$k$ disjoint path cover of a graph $G$ is a collection of pairwise disjoint path subgraphs $P_1,P_2,\dotsc,P_k$ such that each $P_i$ has prescribed vertices $s_i$ and $t_i$ as endpoints and the union of $P_1,P_2,\dotsc,P_k$ contains all vertices of $G$. In this paper, we introduce bipartite transposition-like graphs, which are inductively constructed from lower ranked bipartite transposition-like graphs. We show that every rank $n$ bipartite transposition-like graph $G$ admit a paired $(n-1)$-to-$(n-1)$ disjoint path cover for all choices of $S=\{s_1,s_2,\dotsc,s_{n-1}\}$ and $T=\{t_1,t_2,\dotsc,t_{n-1}\}$, provided that $S$ is in one partite set of $G$ and $T$ is in the other. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11381 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Paired $(n-1)$-to-$(n-1)$ disjoint path covers in bipartite transposition-like graphs Coleman, Anna Fischberg, Gabrielle Gong, Charles Harrington, Joshua Wong, Tony W. H. Combinatorics 05C45, 05C70, 05C75 A paired $k$-to-$k$ disjoint path cover of a graph $G$ is a collection of pairwise disjoint path subgraphs $P_1,P_2,\dotsc,P_k$ such that each $P_i$ has prescribed vertices $s_i$ and $t_i$ as endpoints and the union of $P_1,P_2,\dotsc,P_k$ contains all vertices of $G$. In this paper, we introduce bipartite transposition-like graphs, which are inductively constructed from lower ranked bipartite transposition-like graphs. We show that every rank $n$ bipartite transposition-like graph $G$ admit a paired $(n-1)$-to-$(n-1)$ disjoint path cover for all choices of $S=\{s_1,s_2,\dotsc,s_{n-1}\}$ and $T=\{t_1,t_2,\dotsc,t_{n-1}\}$, provided that $S$ is in one partite set of $G$ and $T$ is in the other. |
| title | Paired $(n-1)$-to-$(n-1)$ disjoint path covers in bipartite transposition-like graphs |
| topic | Combinatorics 05C45, 05C70, 05C75 |
| url | https://arxiv.org/abs/2402.11381 |