Antimagic orientations of disconnected even regular graphs

Abstract

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∈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 2x0+5 vertices with x0=⌈k−(5d+4)2d−2⌉.