Antimagic labelings of a complete graph

Abstract

In $1990$, Hartsfield and Ringel introduced antimagic graphs. Hartsfield and Ringel conjectured that every connected graph (and in particular, a tree) except $K_2$ is antimagic. In $2010$, Hefetz et al.\ raised two questions: Is every orientation of any simple connected undirected graph antimagic? and Given any undirected graph $G$, does there exist an orientation of $G$ which is antimagic? They call such an orientation an {\it antimagic orientation} of $G$. Recently, Bhavale provided an edge labeling for a given graph on $n$ vertices without isolated vertices. In this paper, using the labeling of Bhavale, we prove that a complete graph $K_n$ for $n \geq 3$ is super antimagic as well as totally antimagic total graph. We also prove that there exists an antimagic orientation of $K_n$ for $n \geq 3$.