Antimagic orientations of graphs with given independence number

TL;DR

This paper investigates antimagic orientations of graphs with large or small independence numbers, providing new results that support the conjecture that all connected graphs have such orientations.

Contribution

It extends antimagic orientation results to graphs with independence number at least half or at most four, offering new evidence for the conjecture.

Findings

Proves antimagic orientations exist for graphs with large independence number.

Establishes antimagic orientations for graphs with small independence number (at most four).

Develops new methods potentially useful for resolving the conjecture.

Abstract

Given a digraph $D$ with $m$ arcs and a bijection $\tau: A(D)\rightarrow \{1, 2, \ldots, m\}$, we say $(D, \tau)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and no two vertices in $D$ have the same vertex-sum under $\tau$, where the vertex-sum of a vertex $u$ in $D$ under $\tau$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. Hefetz, M\"{u}tze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs, biregular bipartite graphs, and graphs with large maximum degree. In this paper, we establish more evidence for the aforementioned conjecture by studying antimagic orientations of graphs $G$ with independence number at least $|V(G)|/2$ or at most four. We obtain several results. The method we develop in this paper may shed some light on attacking the aforementioned conjecture.