Arc-disjoint in- and out-branchings in semicomplete split digraphs
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913548905480192 |
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| author | Ai, Jiangdong Hao, Yiming Li, Zhaoxiang Shao, Qi |
| author_facet | Ai, Jiangdong Hao, Yiming Li, Zhaoxiang Shao, Qi |
| contents | An \emph{out-tree (in-tree)} is an oriented tree where every vertex except one, called the \emph{root}, has in-degree (out-degree) one. An \emph{out-branching $B^+_u$ (in-branching $B^-_u$)} of a digraph $D$ is a spanning out-tree (in-tree) rooted at $u$. A \emph{good $(u,v)$-pair} in $D$ is a pair of branchings $B^+_u, B^-_v$ which are arc-disjoint. Thomassen proved that deciding whether a digraph has any good pair is NP-complete. A \emph{semicomplete split digraph} is a digraph where the vertex set is the disjoint union of two non-empty sets, $V_1$ and $V_2$, such that $V_1$ is an independent set, the subdigraph induced by $V_2$ is semicomplete, and every vertex in $V_1$ is adjacent to every vertex in $V_2$. In this paper, we prove that every $2$-arc-strong semicomplete split digraph $D$ contains a good $(u, v)$-pair for any choice of vertices $u, v$ of $D$, thereby confirming a conjecture by Bang-Jensen and Wang [Bang-Jensen and Wang, J. Graph Theory, 2024]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12575 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arc-disjoint in- and out-branchings in semicomplete split digraphs Ai, Jiangdong Hao, Yiming Li, Zhaoxiang Shao, Qi Combinatorics An \emph{out-tree (in-tree)} is an oriented tree where every vertex except one, called the \emph{root}, has in-degree (out-degree) one. An \emph{out-branching $B^+_u$ (in-branching $B^-_u$)} of a digraph $D$ is a spanning out-tree (in-tree) rooted at $u$. A \emph{good $(u,v)$-pair} in $D$ is a pair of branchings $B^+_u, B^-_v$ which are arc-disjoint. Thomassen proved that deciding whether a digraph has any good pair is NP-complete. A \emph{semicomplete split digraph} is a digraph where the vertex set is the disjoint union of two non-empty sets, $V_1$ and $V_2$, such that $V_1$ is an independent set, the subdigraph induced by $V_2$ is semicomplete, and every vertex in $V_1$ is adjacent to every vertex in $V_2$. In this paper, we prove that every $2$-arc-strong semicomplete split digraph $D$ contains a good $(u, v)$-pair for any choice of vertices $u, v$ of $D$, thereby confirming a conjecture by Bang-Jensen and Wang [Bang-Jensen and Wang, J. Graph Theory, 2024]. |
| title | Arc-disjoint in- and out-branchings in semicomplete split digraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.12575 |