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.

中文翻译：

---

具有给定独立数的图的反魔术方向

给定一个有 $m$ 弧和一个双射 $\tau: A(D)\rightarrow \{1, 2, \ldots, m\}$ 的有向图 $D$，我们说 $(D, \tau)$ 是一个图 $G$ 的反魔法方向如果 $D$ 是 $G$ 的方向并且 $D$ 中没有两个顶点在 $\tau$ 下具有相同的顶点和，其中顶点 $u$ 的顶点和在 $\tau$ 下的 $D$ 中是所有进入 $u$ 的弧的标签总和减去所有离开 $u$ 的弧的标签总和。Hefetz, M\"{u}tze, and Schwartz 于 2010 年开始研究图的反魔方向，并推测每个连通图都承认一个反魔方向。这个猜想似乎很难，相关结果鲜为人知。然而，它有被验证对正则图、双正则二部图和最大度数较大的图是正确的。在本文中，我们通过研究独立数至少为 $|V(G)|/2$ 或至多为 4 的图 $G$ 的反幻方向，为上述猜想建立了更多证据。我们得到了几个结果。我们在本文中开发的方法可能会对攻击上述猜想有所启发。