Antimagic orientation of lobsters

Abstract

Let m≥1 be an integer and G be a graph with m edges. We say that G has an antimagic orientation if G has an orientation D and a bijection τ:A(D)→{1,2,…,m} such that no two vertices in D have the same vertex-sum under τ, where the vertex-sum of a vertex u in D under τ is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. Hefetz et al. (2010) conjectured that every connected graph admits an antimagic orientation. The conjecture was confirmed for certain classes of graphs such as dense graphs, regular graphs, and trees including caterpillars and complete k-ary trees. In this note, we prove that every lobster admits an antimagic orientation.