| Title: | On distance magic labelings of Hamming graphs and folded hypercubes |
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| Authors: | ID Miklavič, Štefko (Author)
ID Šparl, Primož (Author) |
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| Files: |  PDF - Presentation file, download (484,67 KB)
MD5: 414A1788ACBCBCE624F787C989A60DA3
 URL - Source URL, visit https://www.dmgt.uz.zgora.pl/publish/article.php?doi=2430
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | Let $\Gamma = (V,E)$ be a graph of order $n$. A distance magic labeling of $\Gamma$ is a bijection $\ell \colon V \to \{1,2, \ldots, n\}$ for which there exists a positive integer $k$ such that $\sum_{x \in N(u)} \ell(x) = k$ for all vertices $u \in V$, where $N(u)$ is the neighborhood of $u$. A graph is said to be distance magic if it admits a distance magic labeling. The Hamming graph $\mathrm{H}(D,q)$, where $D, q$ are positive integers, is the graph whose vertex set consists of all words of length $D$ over an alphabet of size $q$ in which two vertices are adjacent whenever the corresponding words differ in precisely one position. The well-known hypercubes are precisely the Hamming graphs with $q = 2$. Distance magic hypercubes were classified in two papers from 2013 and 2016. In this paper we consider all Hamming graphs. We provide a sufficient condition for a Hamming graph to be distance magic and as a corollary provide an infinite number of pairs $(D, q)$ for which the corresponding Hamming graph $\mathrm{H}(D,q)$ is distance magic. A folded hypercube is a graph obtained from a hypercube by identifying pairs of vertices at maximal distance. We classify distance magic folded hypercubes by showing that the dimension-$D$ folded hypercube is distance magic if and only if $D$ is divisible by $4$. |
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| Keywords: | distance magic labeling, distance magic graph, Hamming graph, folded hypercube |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | str. 17-33 |
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| Numbering: | Vol. 44, iss. 1 |
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| PID: | 20.500.12556/DiRROS-20503  |
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| UDC: | 519.17 |
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| ISSN on article: | 1234-3099 |
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| DOI: | 10.7151/dmgt.2430 |
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| COBISS.SI-ID: | 87398659 |
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| Publication date in DiRROS: | 02.10.2024 |
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| Views: | 16 |
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| Downloads: | 8 |
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