On distance magic circulants of valency 6
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912145993629696 |
|---|---|
| author | Miklavič, Štefko Šparl, Primož |
| author_facet | Miklavič, Štefko Šparl, Primož |
| contents | A graph $Γ= (V,E)$ of order $n$ is {\em distance magic} if it admits a bijective labeling $\ell \colon V \to \{1,2, \ldots, n\}$ of its vertices for which there exists a positive integer $κ$ such that $\sum_{u \in N(v)} \ell(u) = κ$ for all vertices $v \in V$, where $N(v)$ is the neighborhood of $v$. %It is well known that a regular distance magic graph is necessarily of even valency.
A {\em circulant} is a graph admitting an automorphism cyclically permuting its vertices. In this paper we study distance magic circulants of valency $6$. We obtain some necessary and some sufficient conditions for a circulant of valency $6$ to be distance magic, thereby finding several infinite families of examples. The combined results of this paper provide a partial classification of all distance magic circulants of valency $6$. In particular, we classify distance magic circulants of valency $6$, whose order is not divisible by $12$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_09856 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On distance magic circulants of valency 6 Miklavič, Štefko Šparl, Primož Combinatorics 05C78, 05C50, 05C25 A graph $Γ= (V,E)$ of order $n$ is {\em distance magic} if it admits a bijective labeling $\ell \colon V \to \{1,2, \ldots, n\}$ of its vertices for which there exists a positive integer $κ$ such that $\sum_{u \in N(v)} \ell(u) = κ$ for all vertices $v \in V$, where $N(v)$ is the neighborhood of $v$. %It is well known that a regular distance magic graph is necessarily of even valency. A {\em circulant} is a graph admitting an automorphism cyclically permuting its vertices. In this paper we study distance magic circulants of valency $6$. We obtain some necessary and some sufficient conditions for a circulant of valency $6$ to be distance magic, thereby finding several infinite families of examples. The combined results of this paper provide a partial classification of all distance magic circulants of valency $6$. In particular, we classify distance magic circulants of valency $6$, whose order is not divisible by $12$. |
| title | On distance magic circulants of valency 6 |
| topic | Combinatorics 05C78, 05C50, 05C25 |
| url | https://arxiv.org/abs/2203.09856 |