Given a digraph $D$ with $m $ arcs, a bijection $\tau: A(D)\rightarrow \{1, 2, \ldots, m\}$ is an antimagic labeling of $D$ if no two vertices in $D$ have the same vertex-sum, 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 $u$. We say $(D, \tau)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and $\tau$ is an antimagic labeling of $D$. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, Mutze, 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 and biregular bipartite graphs. In this paper, we prove that every connected graph $G$ on $n\ge9$ vertices with maximum degree at least $n-5$ admits an antimagic orientation.

中文翻译：

---

具有大最大度的图的反魔术方向

给定一个有 $m $ 弧的有向图 $D$，如果没有两个顶点，则双射 $\tau: A(D)\rightarrow \{1, 2, \ldots, m\}$ 是 $D$ 的反魔术标签$D$ 具有相同的顶点和，其中 $\tau$ 下的 $D$ 中的顶点 $u $ 的顶点和是输入 $u$ 的所有弧的标签之和减去所有弧的标签之和离开 $u$。如果$D$ 是$G$ 的一个方向并且$\tau$ 是$D$ 的一个反魔术标签，我们说$(D,\tau)$ 是一个图$G$ 的反魔术方向。受 1990 年 Hartsfield 和 Ringel 对图的反魔法标记的猜想的启发，Hefetz、Mutze 和 Schwartz 于 2010 年开始研究图的反魔法方向，并推测每个连通图都承认一个反魔法方向。这个猜想似乎很难，相关结果鲜为人知。然而，它已被验证适用于正则图和双正则二部图。在本文中，我们证明了最大度数至少为 $n-5$ 的 $n\ge9$ 顶点上的每个连通图 $G$ 都承认反幻方向。