On $k$-Shifted Antimagic Spider Forests
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866909298518392832 |
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| author | Chang, Fei-Huang Li, Wei-Tian Liu, Der-Fen Daphne Pan, Zhishi |
| author_facet | Chang, Fei-Huang Li, Wei-Tian Liu, Der-Fen Daphne Pan, Zhishi |
| contents | Let $G(V,E)$ be a simple graph with $m$ edges. For a given integer $k$, a $k$-shifted antimagic labeling is a bijection $f: E(G) \to \{k+1, k+2, \ldots, k+m\}$ such that all vertices have different vertex-sums, where the vertex-sum of a vertex $v$ is the total of the labels assigned to the edges incident to $v$. A graph $G$ is {\it $k$-shifted antimagic} if it admits a $k$-shifted antimagic labeling. For the special case when $k=0$, a $0$-shifted antimagic labeling is known as {\it antimagic labeling}; and $G$ is {\it antimagic} if it admits an antimagic labeling. A spider is a tree with exactly one vertex of degree greater than two. A spider forest is a graph where each component is a spider. In this article, we prove that certain spider forests are $k$-shifted antimagic for all $k \geq 0$. In addition, we show that for a spider forest $G$ with $m$ edges, there exists a positive integer $k_0< m$ such that $G$ is $k$-shifted antimagic for all $k \geq k_0$ and $k \leq -(m+k_0+1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_13582 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On $k$-Shifted Antimagic Spider Forests Chang, Fei-Huang Li, Wei-Tian Liu, Der-Fen Daphne Pan, Zhishi Combinatorics 05C78 Let $G(V,E)$ be a simple graph with $m$ edges. For a given integer $k$, a $k$-shifted antimagic labeling is a bijection $f: E(G) \to \{k+1, k+2, \ldots, k+m\}$ such that all vertices have different vertex-sums, where the vertex-sum of a vertex $v$ is the total of the labels assigned to the edges incident to $v$. A graph $G$ is {\it $k$-shifted antimagic} if it admits a $k$-shifted antimagic labeling. For the special case when $k=0$, a $0$-shifted antimagic labeling is known as {\it antimagic labeling}; and $G$ is {\it antimagic} if it admits an antimagic labeling. A spider is a tree with exactly one vertex of degree greater than two. A spider forest is a graph where each component is a spider. In this article, we prove that certain spider forests are $k$-shifted antimagic for all $k \geq 0$. In addition, we show that for a spider forest $G$ with $m$ edges, there exists a positive integer $k_0< m$ such that $G$ is $k$-shifted antimagic for all $k \geq k_0$ and $k \leq -(m+k_0+1)$. |
| title | On $k$-Shifted Antimagic Spider Forests |
| topic | Combinatorics 05C78 |
| url | https://arxiv.org/abs/2112.13582 |