Antimagic orientations of disconnected even regular graphs

A labeling of a digraph D with m arcs is a bijection from the set of arcs of D to {1, 2,..., m}. A labeling of D is antimagic if no two vertices in D have the same vertex sum, where the vertex-sum of a vertex u is an element of V(D) for a labeling is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. An orientation D of a graph G is antimagic if D has an antimagic labeling. Hefetz et al. (2010) raised the question: Does every graph admit an antimagic orientation? It had been proved that every 2d-regular graph with at most two odd components has an antimagic orientation. In this paper, we consider 2d-regular graphs with more than two odd components. We show that every 2d-regular graph with k (3 <= k <= 5d + 4) odd components has an antimagic orientation. And we show that each 2d-regular graph with k (k >= 5d + 5) odd components admits an antimagic orientation if each odd component has at least 2x(0) + 5 vertices with x(0) = right perpendiculark-2(5d+4)/2d-2left perpendicular. (C) 2019 Elsevier B.V. All rights reserved.