A generalization of the subdivision of antimagic trees with at most one vertex of degree-2

Kuo, Kai-Ning; Li, Wei-Tian; Liu, Yi-Tsen

doi:10.2140/involve.2025.18.821

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A generalization of the subdivision of antimagic trees with at most one vertex of degree-2

Kai-Ning Kuo, Wei-Tian Li and Yi-Tsen Liu

Vol. 18 (2025), No. 5, 821–828

DOI: 10.2140/involve.2025.18.821

Abstract

A bijection between the edge set of a graph $G$ and the set ${1, 2, \dots, m}$ , where $m$ is the number of edges of $G$ , is called an antimagic labeling for $G$ if when we sum up the labels of the edges incident to a vertex, different vertices have different sums. Hartsfield and Ringel conjectured that every tree other than an edge has an antimagic labeling. Liang, Wong, and Zhu proved that every tree contains at most one vertex of degree-2 is antimagic. Moreover, if a tree contains no vertex of degree-2, then subdividing every edge once yields a new tree that is also antimagic. In this paper, we prove that for any tree $T$ with at most one vertex of degree-2, the tree obtained by subdividing all edges of $T$ the same number of times is antimagic.

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Keywords

antimagic labeling, subdivision, tree

Mathematical Subject Classification

Primary: 05C78

Milestones

Received: 9 November 2023

Revised: 26 April 2024

Accepted: 27 April 2024

Published: 20 November 2025

Communicated by Glenn Hurlbert

Authors

Kai-Ning Kuo
Taichung Municipal Taichung Girls’ Senior High School Taichung Taiwan

Wei-Tian Li
Department of Applied Mathematics National Chung Hsing University Taichung Taiwan

Yi-Tsen Liu
Taichung Municipal Taichung Girls’ Senior High School Taichung Taiwan

Vol. 18, No. 5, 2025