Sufficient conditions for closed-trailable in digraphs

Abstract

A digraph $D$ with a subset $S$ of $V(D)$ is called $\mathbf{S}${\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 $\mathbf{S}${\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 $d(u)+d(v)\ge 2n-3$ 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 $\hat{I}{´}^0(D⟨S⟩)\ge \hat{I}{\pm }^{\prime }(D⟨S⟩)>0$, where $\hat{I}{´}^0(D⟨S⟩)$ is the minimum semi-degree of $D⟨S⟩$ and $\hat{I}{\pm }^{\prime }(D⟨S⟩)$ is the matching number of $D⟨S⟩$, then $S$ is closed-trailable. This result generalizes the theorem of Algefari et al. \cite{AlLa15} on supereulerianity.

13 pages