Shifted-Antimagic Labelings for Graphs

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