Local antimagic orientations of [formula omitted]-degenerate graphs

Abstract

A k-antimagic labeling of a digraph D with n vertices and m arcs is an injection from the set of arcs of D to the integers {1,…,m+k} such that all n vertex-sums are pairwise distinct, where the vertex-sum of a vertex v is the sum of labels of all arcs entering v minus the sum of labels of all arcs leaving v. An orientation D of a graph G is called k-antimagic if D has a k-antimagic labeling. Hefetz et al. (2010) conjectured that every connected graph admits an antimagic orientation, where “antimagic” is short for “0-antimagic”. In this paper, we consider local k-antimagic orientations of graphs. An orientation D of a graph G is called local k-antimagic if there is an injective edge labeling from E(G) to {1,…,|E(G)|+k} such that any two adjacent vertices of D have different vertex-sums. We prove that every d-degenerate graph admits a local (d+2)-antimagic orientation.