Dominated Pair Degree Sum Conditions of Supereulerian Digraphs

Abstract

Abstract A digraph D is supereulerian if D contains a spanning eulerian subdigraph. In this paper, we propose the following problem: is there an integer t with 0 ≤ t ≤ n − 3 so that any strong digraph with n vertices satisfying either both d(u) ≥ n − 1 + t and d(v) ≥ n − 2 − t or both d(u) ≥ n − 2 − t and d(v) ≥ n − 1 + t, for any pair of dominated or dominating nonadjacent vertices {u, v}, is supereulerian? We prove the cases when t = 0, t = n − 4 and t = n − 3. Moreover, we show that if a strong digraph D with n vertices satisfies min{d+(u)+d−(v), d−(u)+d+(v)} ≥ n−1 for any pair of dominated or dominating nonadjacent vertices {u, v} of D, then D is supereulerian.