Local antimagic orientation of graphs

Yulin Chang (), Fei Jing () and Guanghui Wang ()
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Yulin Chang: Shandong University
Fei Jing: Shandong University
Guanghui Wang: Shandong University

Journal of Combinatorial Optimization, 2020, vol. 39, issue 4, No 9, 1129-1152

Abstract: Abstract An antimagic labelling of a digraph D with m arcs is a bijection from the set of arcs of D to $$\{1,\ldots ,m\}$${1,…,m} such that any two vertices have distinct vertex-sums, where the vertex-sum of a vertex $$v\in V(D)$$v∈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$$\ddot{\text {u}}$$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 $$\{1,\ldots ,|E(G)|\}$${1,…,|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.

Keywords: Antimagic orientation; Local antimagic orientation; Combinatorial Nullstellensatz (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10878-020-00551-x

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