Antimagic Labeling for Unions of Graphs with Many Three-Paths

TL;DR

This paper investigates the antimagic labeling of graphs, establishing an upper bound on the parameter (G) for all graphs and demonstrating its tightness for various graph families, thereby generalizing previous specific results.

Contribution

It provides a universal upper bound on (G) applicable to all graphs, extending prior work on special graph families and confirming the bound's tightness across multiple cases.

Findings

Established a general upper bound on (G) for all graphs.

Proved the bound is tight for star forests, double stars, and other graph families.

Extended previous results by generalizing bounds to broader classes of graphs.

Abstract

Let $G$ be a graph with $m$ edges and let $f$ be a bijection from $E(G)$ to $\{1,2, \dots, m\}$. For any vertex $v$, denote by $\phi_f(v)$ the sum of $f(e)$ over all edges $e$ incident to $v$. If $\phi_f(v) \neq \phi_f(u)$ holds for any two distinct vertices $u$ and $v$, then $f$ is called an {\it antimagic labeling} of $G$. We call $G$ {\it antimagic} if such a labeling exists. Hartsfield and Ringel in 1991 conjectured that all connected graphs except $P_2$ are antimagic. Denote the disjoint union of graphs $G$ and $H$ by $G \cup H$, and the disjoint union of $t$ copies of $G$ by $tG$. For an antimagic graph $G$ (connected or disconnected), we define the parameter $\tau(G)$ to be the maximum integer such that $G \cup tP_3$ is antimagic for all $t \leq \tau(G)$. Chang, Chen, Li, and Pan showed that for all antimagic graphs $G$, $\tau(G)$ is finite [Graphs and Combinatorics 37 (2021), 1065--1182]. Further, Shang, Lin, Liaw [Util. Math. 97 (2015), 373--385] and Li [Master Thesis, National Chung Hsing University, Taiwan, 2019] found the exact value of $\tau(G)$ for special families of graphs: star forests and balanced double stars respectively. They did this by finding explicit antimagic labelings of $G\cup tP_3$ and proving a tight upper bound on $\tau(G)$ for these special families. In the present paper, we generalize their results by proving an upper bound on $\tau(G)$ for all graphs. For star forests and balanced double stars, this general bound is equivalent to the bounds given in \cite{star forest} and \cite{double star} and tight. In addition, we prove that the general bound is also tight for every other graph we have studied, including an infinite family of jellyfish graphs, cycles $C_n$ where $3 \leq n \leq 9$, and the double triangle $2C_3$.