Sufficient conditions for closed-trailable in digraphs

A digraph D with a subset S of $V(D)$ is called S{ -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 S{ -strictly strong} if there exist two nonadjacent vertices u,v∈ S such that D contains a closed ditrail through the vertices u and v; and define a subset S⊆ V(D) to be { 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)≥ 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. {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 δ⁰(D S)≥α'(D S)>0, where δ⁰(D S) is the minimum semi-degree of D S and α'(D S) is the matching number of D S, then S is closed-trailable. This result generalizes the theorem of Algefari et al. {AlLa15} on supereulerianity.