Sufficient conditions for closed-trailable in digraphs

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.