Local antimagic orientation of graphs

Abstract

An antimagic labelling of a digraph D with m arcs is a bijection from the set of arcs of D to $\{1,\ldots ,m\}$ such that any two vertices have distinct vertex-sums, where the vertex-sum of a vertex $v\in V(D)$ 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 antimagic if D has an antimagic labelling. In 2010, Hefetz, M$\ddot{\text {u}}$tze and Schwartz conjectured that every connected graph admits an antimagic orientation. The conjecture is still open, even for trees. Motivated by directed version of the well-known 1-2-3 Conjecture, we deal with vertex-sums such that only adjacent vertices must be distinguished. An orientation D of a graph G is local antimagic if there is a bijection from E(G) to $\{1,\ldots ,|E(G)|\}$ such that any two adjacent vertices have distinct vertex-sums. We prove that every graph with maximum degree at most 4 admits a local antimagic orientation by Alon’s Combinatorial Nullstellensatz.