Local antimagic orientation of graphs

An antimagic labelling of a digraph D with m arcs is a bijection from the set of arcs of D to {1,…,m}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1,\ldots ,m\}$$\end{document} such that any two vertices have distinct vertex-sums, where the vertex-sum of a vertex v∈V(D)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v\in V(D)$$\end{document} 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, Mu¨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\text {u}}$$\end{document}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,…,|E(G)|}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1,\ldots ,|E(G)|\}$$\end{document} 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.