Antimagic labelings of a complete graph

TL;DR

This paper proves that complete graphs with at least three vertices are super antimagic and totally antimagic, and also establishes the existence of an antimagic orientation for such graphs, advancing the understanding of antimagic labelings.

Contribution

It demonstrates that complete graphs are super antimagic and totally antimagic, and confirms the existence of antimagic orientations for all complete graphs with at least three vertices.

Findings

Complete graphs $K_n$ are super antimagic.

Complete graphs $K_n$ are totally antimagic.

Existence of antimagic orientations for $K_n$ with $n \\geq 3$.

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 $n \geq 3$.