Antimagic orientations of graphs with large maximum degree

Given a digraph D with m arcs, a bijection tau : A(D) -> {1, 2, ... , m} is an antimagic labeling of D if no two vertices in D have the same vertex-sum, where the vertex -sum of a vertex u in D under tau is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. We say (D, tau) is an antimagic orientation of a graph G if D is an orientation of G and tau is an antimagic labeling of D. Motivated by the conjecture of Hartsfield and Ringel (1990) on antimagic labelings of graphs, Hefetz et al. (2010) initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs and biregular bipartite graphs. In this paper, we prove that every connected graph G on n >= 9 vertices with maximum degree at least n - 5 admits an antimagic orientation. (c) 2020 Elsevier B.V. All rights reserved.