On the 2-Linkage Problem for Split Digraphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915843392143360 |
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| author | Chen, Xiaoying Bang-Jensen, Jørgen Yan, Jin Zhou, Jia |
| author_facet | Chen, Xiaoying Bang-Jensen, Jørgen Yan, Jin Zhou, Jia |
| contents | A digraph is {\bf \( k \)-linked} if for arbitary two disjoint vertex sets \(\{s_1, \ldots, s_k\}\) and \(\{t_1, \ldots, t_k\}\), there exist vertex-disjoint directed paths \(P_1, \ldots, P_k\) {such that \(P_i\) is a directed path from \(s_i\) to \(t_i\) for each $i\in [k]$}. A {\bf split digraph} is a digraph \( D = (V_1, V_2; A) \) whose vertex set is a disjoint union of two nonempty sets \( V_1 \) and \( V_2 \) such that \( V_1 \) is an independent set and the subdigraph induced by \( V_2 \) is semicomplete (no pair of non-adjacent vertices). A {\bf semicomplete split digraph} is a split digraph \( D = (V_1, V_2; A) \) in which every vertex in the independent set \( V_1 \) is adjacent to every vertex in \( V_2 \). {Semicomplete split digraphs form an important subclass of the class of semicomplete multipartite digraphs.} In this paper, we prove that every 6-strong split digraph is 2-linked. This solves a problem posed by Bang-Jensen and Wang [J. Graph Theory, 2025]. We also show that every 5-strong semicomplete split digraph is 2-linked. This bound is tight already for semicomplete digraphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_07603 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the 2-Linkage Problem for Split Digraphs Chen, Xiaoying Bang-Jensen, Jørgen Yan, Jin Zhou, Jia Combinatorics 05C20, 05C38, 05C40 A digraph is {\bf \( k \)-linked} if for arbitary two disjoint vertex sets \(\{s_1, \ldots, s_k\}\) and \(\{t_1, \ldots, t_k\}\), there exist vertex-disjoint directed paths \(P_1, \ldots, P_k\) {such that \(P_i\) is a directed path from \(s_i\) to \(t_i\) for each $i\in [k]$}. A {\bf split digraph} is a digraph \( D = (V_1, V_2; A) \) whose vertex set is a disjoint union of two nonempty sets \( V_1 \) and \( V_2 \) such that \( V_1 \) is an independent set and the subdigraph induced by \( V_2 \) is semicomplete (no pair of non-adjacent vertices). A {\bf semicomplete split digraph} is a split digraph \( D = (V_1, V_2; A) \) in which every vertex in the independent set \( V_1 \) is adjacent to every vertex in \( V_2 \). {Semicomplete split digraphs form an important subclass of the class of semicomplete multipartite digraphs.} In this paper, we prove that every 6-strong split digraph is 2-linked. This solves a problem posed by Bang-Jensen and Wang [J. Graph Theory, 2025]. We also show that every 5-strong semicomplete split digraph is 2-linked. This bound is tight already for semicomplete digraphs. |
| title | On the 2-Linkage Problem for Split Digraphs |
| topic | Combinatorics 05C20, 05C38, 05C40 |
| url | https://arxiv.org/abs/2603.07603 |