Every connected graph admits a local antimagic orientation and almost every graph admits an antimagic orientation

Abstract

An undirected graph $G$ is said to admit an antimagic orientation if there exist an orientation $D$ and a bijection between $E(G)$ and $\{1,2,\ldots,|E(G)|\}$ such that any two vertices have distinct vertex sums, where the vertex sum of a vertex is the sum of the labels of the in-edges minus that of the out-edges incident to the vertex. It is conjectured by Hefetz, M\"{u}tze, and Schwartz that every connected graph admits an antimagic orientation. A weak version of this problem is to require the distinct vertex sums only for the adjacent vertices. In that case, we say the graph admits a local antimagic orientation. Chang, Jing, and Wang~\cite{CJW20} conjectured that every connected graph admits a local antimagic orientation. In this paper, we give an affirmative answer to the conjecture of Chang et al., and show that almost every graph satisfies the conjecture of Hefetz et al.