Antimagic and product antimagic graphs with pendant edges

TL;DR

This paper proves that certain connected graphs with support vertices are antimagic and product antimagic for specific label sets, supporting the longstanding conjecture that all such graphs are antimagic.

Contribution

It establishes that connected graphs with support vertices are antimagic for any arithmetic sequence of labels and also admits product antimagic labelings under certain conditions.

Findings

Graphs with support vertices are antimagic for arithmetic sequences.

Such graphs are also product antimagic if the smallest label is at least one.

The proofs are constructive, providing explicit labelings.

Abstract

Let $G=(V,E)$ be a simple graph of size $m$ and $L$ a set of $m$ distinct real numbers. An $L$-labeling of $G$ is a bijection $\phi: E \rightarrow L$. We say that $\phi$ is an antimagic $L$-labeling if the induced vertex sum $\phi_+: V \rightarrow \mathbb {R}$ defined as $\phi_+(u)=\sum_{uv\in E}\phi(uv)$ is injective. Similarly, $\phi$ is a product antimagic $L$-labeling of $G$ if the induced vertex product $\phi_{\circ}: V \rightarrow \mathbb {R}$ defined as $\phi_{\circ}(u)=\prod_{uv\in E}\phi(uv)$ is injective. A graph $G$ is antimagic (resp. product antimagic) if it has an antimagic (resp. a product antimagic) $L$-labeling for $L=\{1,2,\dots,m\}$. Hartsfield and Ringel conjectured that every simple connected graph distinct from $K_2$ is antimagic, but the conjecture remains widely open. We prove, among other results, that every connected graph of size $m$, $m \geq 3$, admits an antimagic $L$-labeling for every arithmetic sequence $L$ of $m$ positive real numbers, if every vertex of degree at least three is a support vertex. As a corollary, we derive that these graphs are antimagic, reinforcing the veracity of the conjecture by Hartsfield and Ringel. Moreover, these graphs admit also a product antimagic $L$-labeling provided that the smallest element of $L$ is at least one. The proof is constructive.