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.