Chromatic number of graphs with special distance sets, I
Given a subset \(D\) of positive integers, an integer distance graph is a graph \(G(\mathbb{Z}, D)\) with the set \(\mathbb{Z}\) of integers as vertex set and with an edge joining two vertices \(u\) and \(v\) if and only if \(|u - v| \in D\). In this paper we consider the problem of determining the...
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| Дата: | 2018 |
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| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-10282018-04-26T01:41:11Z Chromatic number of graphs with special distance sets, I Yegnanarayanan, V. chromatic number, prime distance graph, unit distance graph 05C15 Given a subset \(D\) of positive integers, an integer distance graph is a graph \(G(\mathbb{Z}, D)\) with the set \(\mathbb{Z}\) of integers as vertex set and with an edge joining two vertices \(u\) and \(v\) if and only if \(|u - v| \in D\). In this paper we consider the problem of determining the chromatic number of certain integer distance graphs \(G(\mathbb{Z}, D)\)whose distance set \(D\) is either 1) a set of \((n+1)\) positive integers for which the \(n^{th}\) power of the last is the sum of the \(n^{th}\) powers of the previous terms, or 2) a set of pythagorean quadruples, or 3) a set of pythagorean \(n\)-tuples, or 4) a set of square distances, or 5) a set of abundant numbers or deficient numbers or carmichael numbers, or 6) a set of polytopic numbers, or 7) a set of happy numbers or lucky numbers, or 8) a set of Lucas numbers, or 9) a set of \(\mathcal{U}\)lam numbers, or 10) a set of weird numbers. Besides finding the chromatic number of a few specific distance graphs we also give useful upper and lower bounds for general cases. Further, we raise some open problems. Lugansk National Taras Shevchenko University 2018-04-26 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1028 Algebra and Discrete Mathematics; Vol 17, No 1 (2014) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1028/552 Copyright (c) 2018 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
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| datestamp_date |
2018-04-26T01:41:11Z |
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OJS |
| language |
English |
| topic |
chromatic number prime distance graph unit distance graph 05C15 |
| spellingShingle |
chromatic number prime distance graph unit distance graph 05C15 Yegnanarayanan, V. Chromatic number of graphs with special distance sets, I |
| topic_facet |
chromatic number prime distance graph unit distance graph 05C15 |
| format |
Article |
| author |
Yegnanarayanan, V. |
| author_facet |
Yegnanarayanan, V. |
| author_sort |
Yegnanarayanan, V. |
| title |
Chromatic number of graphs with special distance sets, I |
| title_short |
Chromatic number of graphs with special distance sets, I |
| title_full |
Chromatic number of graphs with special distance sets, I |
| title_fullStr |
Chromatic number of graphs with special distance sets, I |
| title_full_unstemmed |
Chromatic number of graphs with special distance sets, I |
| title_sort |
chromatic number of graphs with special distance sets, i |
| description |
Given a subset \(D\) of positive integers, an integer distance graph is a graph \(G(\mathbb{Z}, D)\) with the set \(\mathbb{Z}\) of integers as vertex set and with an edge joining two vertices \(u\) and \(v\) if and only if \(|u - v| \in D\). In this paper we consider the problem of determining the chromatic number of certain integer distance graphs \(G(\mathbb{Z}, D)\)whose distance set \(D\) is either 1) a set of \((n+1)\) positive integers for which the \(n^{th}\) power of the last is the sum of the \(n^{th}\) powers of the previous terms, or 2) a set of pythagorean quadruples, or 3) a set of pythagorean \(n\)-tuples, or 4) a set of square distances, or 5) a set of abundant numbers or deficient numbers or carmichael numbers, or 6) a set of polytopic numbers, or 7) a set of happy numbers or lucky numbers, or 8) a set of Lucas numbers, or 9) a set of \(\mathcal{U}\)lam numbers, or 10) a set of weird numbers. Besides finding the chromatic number of a few specific distance graphs we also give useful upper and lower bounds for general cases. Further, we raise some open problems. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1028 |
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AT yegnanarayananv chromaticnumberofgraphswithspecialdistancesetsi |
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2025-07-17T10:30:48Z |
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2025-07-17T10:30:48Z |
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