The concept of antimagic labelings of a graph is to produce distinct vertex sums by labeling edges through consecutive numbers starting from one. A long-standing conjecture is that every connected graph, except a single edge, is antimagic. Some graphs are known to be antimagic, but little has been known about sparse graphs, not even trees. This paper studies a weak version called k-shifted-antimagic labelings which allow the consecutive numbers starting from \(k+1\), instead of starting from 1, where k can be any integer. This paper establishes connections among various concepts proposed in the literature of antimagic labelings and extends previous results in three aspects:

• Some classes of graphs, including trees and graphs whose vertices are of odd degrees, which have not been verified to be antimagic are shown to be k-shifted-antimagic for sufficiently large k.

• Some graphs are proved k-shifted-antimagic for all k, while some are proved not for some particular k.

• Disconnected graphs are also considered.

图的反魔术标记的概念是通过以从1开始的连续数字标记边来产生不同的顶点和。一个长期的推测是，每个连接的图形（除单个边沿之外）都是反魔术的。某些图被认为是反魔术的，但是对于稀疏图却知之甚少，甚至树木也鲜为人知。本文研究了一种弱的版本，称为k位移反魔术标签，它允许从\（k + 1 \）开始的连续数字，而不是从1开始的连续数字，其中k可以是任何整数。本文建立了抗磁性标签文献中提出的各种概念之间的联系，并将先前的结果扩展到三个方面：

• 对于足够大的k，某些类别的图（包括树和顶点为奇数度的图）尚未经过验证是反魔术的，它们显示为k位移反魔术的。

• 对于所有k，有些图被证明是k位移反魔术的，而对于某些特定的k，有些图被证明不是k。

• 还考虑了断开连接的图。