Shifted-antimagic Labelings for Graphs

TL;DR

This paper introduces and studies $k$-shifted-antimagic labelings, a generalization of antimagic labelings, establishing new results for various classes of graphs including trees, graphs with odd degrees, and disconnected graphs.

Contribution

It extends antimagic labeling theory by analyzing $k$-shifted variants, proving new results for classes of graphs previously not verified, and exploring the behavior across different $k$ values.

Findings

Certain trees and graphs with odd degrees are $k$-shifted-antimagic for large $k$

Some graphs are $k$-shifted-antimagic for all $k$

Disconnected graphs are also analyzed in the context of $k$-shifted-antimagic labelings

Abstract

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: $\bullet$ 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 $k$. $\bullet$ Some graphs are proved $k$-shifted-antimagic for all $k$, while some are proved not for some particular $k$. $\bullet$ Disconnected graphs are also considered.