Sufficient conditions for closed-trailable in digraphs
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
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2024
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| _version_ | 1866910495608406016 |
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| author | Dong, Changchang Yang, Hong Meng, Jixiang Liu, Juan |
| author_facet | Dong, Changchang Yang, Hong Meng, Jixiang Liu, Juan |
| contents | A digraph $D$ with a subset $S$ of $V(D)$ is called $\boldsymbol{S}${\bf -strong} if for every pair of distinct vertices $u$ and $v$ of $S$, there is a $(u, v)$-dipath and a $(v, u)$-dipath in $D$. We define a digraph $D$ with a subset $S$ of $V(D)$ to be $\boldsymbol{S}${\bf -strictly strong} if there exist two nonadjacent vertices $u,v\in S$ such that $D$ contains a closed ditrail through the vertices $u$ and $v$; and define a subset $S\subseteq V(D)$ to be {\bf closed-trailable} if $D$ contains a closed ditrail through all the vertices of $S$. In this paper, we prove that for a digraph $D$ with $n$ vertices and a subset $S$ of $V(D)$, if $D$ is $S$-strong and if $d(u) + d(v)\geq 2n -3$ for any two nonadjacent vertices $u,v$ of $S$, then $S$ is closed-trailable. This result generalizes the theorem of Bang-Jensen et al. \cite{BaMa14} on supereulerianity. Moveover, we show that for a digraph $D$ and a subset $S$ of $V(D)$, if $D$ is $S$-strictly strong and if $δ^0(D\langle S\rangle)\geqα'(D\langle S\rangle)>0$, where $δ^0(D\langle S\rangle)$ is the minimum semi-degree of $D\langle S\rangle$ and $α'(D\langle S\rangle)$ is the matching number of $D\langle S\rangle$, then $S$ is closed-trailable. This result generalizes the theorem of Algefari et al. \cite{AlLa15} on supereulerianity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14332 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sufficient conditions for closed-trailable in digraphs Dong, Changchang Yang, Hong Meng, Jixiang Liu, Juan Combinatorics A digraph $D$ with a subset $S$ of $V(D)$ is called $\boldsymbol{S}${\bf -strong} if for every pair of distinct vertices $u$ and $v$ of $S$, there is a $(u, v)$-dipath and a $(v, u)$-dipath in $D$. We define a digraph $D$ with a subset $S$ of $V(D)$ to be $\boldsymbol{S}${\bf -strictly strong} if there exist two nonadjacent vertices $u,v\in S$ such that $D$ contains a closed ditrail through the vertices $u$ and $v$; and define a subset $S\subseteq V(D)$ to be {\bf closed-trailable} if $D$ contains a closed ditrail through all the vertices of $S$. In this paper, we prove that for a digraph $D$ with $n$ vertices and a subset $S$ of $V(D)$, if $D$ is $S$-strong and if $d(u) + d(v)\geq 2n -3$ for any two nonadjacent vertices $u,v$ of $S$, then $S$ is closed-trailable. This result generalizes the theorem of Bang-Jensen et al. \cite{BaMa14} on supereulerianity. Moveover, we show that for a digraph $D$ and a subset $S$ of $V(D)$, if $D$ is $S$-strictly strong and if $δ^0(D\langle S\rangle)\geqα'(D\langle S\rangle)>0$, where $δ^0(D\langle S\rangle)$ is the minimum semi-degree of $D\langle S\rangle$ and $α'(D\langle S\rangle)$ is the matching number of $D\langle S\rangle$, then $S$ is closed-trailable. This result generalizes the theorem of Algefari et al. \cite{AlLa15} on supereulerianity. |
| title | Sufficient conditions for closed-trailable in digraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.14332 |