Antimagic and product antimagic graphs with pendant edges
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| Format: | Preprint |
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2024
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| _version_ | 1866914789124472832 |
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| author | Mora, Mercè Tey, Joaquín |
| author_facet | Mora, Mercè Tey, Joaquín |
| contents | 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 $ϕ: E \rightarrow L$. We say that $ϕ$ is an antimagic $L$-labeling if the induced vertex sum $ϕ_+: V \rightarrow \mathbb {R}$ defined as $ϕ_+(u)=\sum_{uv\in E}ϕ(uv)$ is injective. Similarly, $ϕ$ is a product antimagic $L$-labeling of $G$ if the induced vertex product $ϕ_{\circ}: V \rightarrow \mathbb {R}$ defined as $ϕ_{\circ}(u)=\prod_{uv\in E}ϕ(uv)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_05375 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Antimagic and product antimagic graphs with pendant edges Mora, Mercè Tey, Joaquín Combinatorics 05C78 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 $ϕ: E \rightarrow L$. We say that $ϕ$ is an antimagic $L$-labeling if the induced vertex sum $ϕ_+: V \rightarrow \mathbb {R}$ defined as $ϕ_+(u)=\sum_{uv\in E}ϕ(uv)$ is injective. Similarly, $ϕ$ is a product antimagic $L$-labeling of $G$ if the induced vertex product $ϕ_{\circ}: V \rightarrow \mathbb {R}$ defined as $ϕ_{\circ}(u)=\prod_{uv\in E}ϕ(uv)$ 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. |
| title | Antimagic and product antimagic graphs with pendant edges |
| topic | Combinatorics 05C78 |
| url | https://arxiv.org/abs/2405.05375 |