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

TL;DR

This paper establishes a new degree sum condition involving dominated and dominating pairs in digraphs, providing criteria for a digraph to be supereulerian, extending previous results on degree conditions for such graphs.

Contribution

It introduces a novel condition on dominated and dominating pair degree sums that guarantees a digraph is supereulerian, broadening the understanding of degree conditions for Eulerian properties.

Findings

New degree sum condition for dominated/dominating pairs in supereulerian digraphs

Extension of previous degree conditions to broader classes of vertex pairs

Characterization of when a digraph is supereulerian based on these conditions

Abstract

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 $d(u) + d(v)\geq 2n -3$ 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.