Antimagic orientations of disconnected even regular graphs
Song Chen, Rong‐Xia Hao · Discrete Mathematics · 2019
A l a b e l i n g 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 a n t i m a g i c 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 a n t i m a g i c 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 2 d -regular graph with at most two odd components has an antimagic orientation. In this paper, we consider 2 d -regular graphs with more than two odd components. We show that every 2 d -regular graph with k ( 3 ≤ k ≤ 5 d + 4 ) odd components has an antimagic orientation. And we show that each 2 d -regular graph with k ( k ≥ 5 d + 5 ) odd components admits an antimagic orientation if each odd component has at least 2 x 0 + 5 vertices with x 0 = ⌈ k − ( 5 d + 4 ) 2 d − 2 ⌉ .