A new condition on dominated pair degree sum for a digraph to be supereulerian

A digraph D is supereulerian if D contains a spanning eulerian subdigraph. For any two vertices $u,v$ in a digraph D, if (u,w),(v,w)∈ A(D) for some w∈ V(D), then we call the pair , v\ dominating; if (w,u),(w,v)∈ A(D) for some w∈ V(D), then we call the pair , v\ dominated. In 2015, Bang-Jensen and Maddaloni [Journal of graph theory, 79(1) (2015) 8-20] proved that if a strong digraph D with n vertices satisfies d(u) + d(v)≥ 2n -3 for any pair of nonadjacent vertices ,v\ of D, then D is supereulerian. In this paper, we study the above degree sum condition for any pair of dominated or dominating nonadjacent vertices of supereulerian digraphs.