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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| Date: | 2018 |
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| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2018
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1028 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543330351972352 |
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| author | Yegnanarayanan, V. |
| author_facet | Yegnanarayanan, V. |
| author_sort | Yegnanarayanan, V. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-04-26T01:41:11Z |
| 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. |
| first_indexed | 2026-02-08T07:57:31Z |
| format | Article |
| id | admjournalluguniveduua-article-1028 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:57:31Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-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 |
| spellingShingle | chromatic number prime distance graph unit distance graph 05C15 Yegnanarayanan, V. Chromatic number of graphs with special distance sets, I |
| title | 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_short | Chromatic number of graphs with special distance sets, I |
| title_sort | chromatic number of graphs with special distance sets, i |
| topic | chromatic number prime distance graph unit distance graph 05C15 |
| topic_facet | chromatic number prime distance graph unit distance graph 05C15 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1028 |
| work_keys_str_mv | AT yegnanarayananv chromaticnumberofgraphswithspecialdistancesetsi |