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 $(u,w),(v,w)\in A\left(D\right)$ for some $w\in V\left(D\right)$, then we call the pair $\{u,v\}$ dominating; if $(w,u),(w,v)\in A\left(D\right)$ for some $w\in V\left(D\right)$, then we call the pair $\{u,v\}$ dominated. In 2015, Bang–Jensen and Maddaloni (2015) proved that if a strong digraph $D$ with $n$ vertices satisfies $d\left(u\right)+d\left(v\right)\ge 2n-3$ for any pair of nonadjacent vertices $\{u,v\}$ of $D$, then $D$ is supereulerian. In this paper, we study degree sum conditions only for any pair of dominated or dominating nonadjacent vertices to assure the digraph to be supereulerian, which imply the above-mentioned result.