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 $(u,w),(v,w)\in A(D)$ for some $w\in V(D)$, then we call the pair $\{u,v\}$ dominating; if $(w,u),(w,v)\in A(D)$ for some $w\in V(D)$, then we call the pair $\{u,v\}$ 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 $d(u)+d(v)\ge 2n-3$ for any pair of nonadjacent vertices $\{u,v\}$ 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.