Antimagic orientations for the complete k-ary trees

Abstract

A labeling of a digraph D with m arcs is a bijection from the set of arcs of D to \(\{1,2,\ldots ,m\}\). A labeling of D is antimagic if all vertex-sums of vertices in D are pairwise distinct, where the vertex-sum of a vertex \(u \in V(D)\) for a labeling is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. Hefetz et al. (J Graph Theory 64:219–232, 2010) conjectured that every connected graph admits an antimagic orientation. We support this conjecture for the complete k-ary trees and show that all the complete k-ary trees \(T_k^r\) with height r have antimagic orientations for any k and r.