Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
For a partially ordered set (A,≤)(A,\le ), let GA{G}_{A} be the simple, undirected graph with vertex set A such that two vertices a≠b∈Aa\ne b\in A are adjacent if either a≤ba\le b or b≤ab\le a. We call GA{G}_{A} the partial order graph or comparability graph of A. Furthermore, we say that a graph G...
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De Gruyter
2020-12-01
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| Series: | Open Mathematics |
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| Online Access: | http://www.degruyter.com/view/j/math.2020.18.issue-1/math-2020-0085/math-2020-0085.xml?format=INT |
| _version_ | 1818418990799650816 |
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| author | Badawi Ayman Rissner Roswitha |
| author_facet | Badawi Ayman Rissner Roswitha |
| author_sort | Badawi Ayman |
| collection | DOAJ |
| description | For a partially ordered set (A,≤)(A,\le ), let GA{G}_{A} be the simple, undirected graph with vertex set A such that two vertices a≠b∈Aa\ne b\in A are adjacent if either a≤ba\le b or b≤ab\le a. We call GA{G}_{A} the partial order graph or comparability graph of A. Furthermore, we say that a graph G is a partial order graph if there exists a partially ordered set A such that G=GAG={G}_{A}. For a class C{\mathcal{C}} of simple, undirected graphs and n, m≥1m\ge 1, we define the Ramsey number
ℛC(n,m){ {\mathcal R} }_{{\mathcal{C}}}(n,m) with respect to C{\mathcal{C}} to be the minimal number of vertices r such that every induced subgraph of an arbitrary graph in C{\mathcal{C}} consisting of r vertices contains either a complete n-clique Kn{K}_{n} or an independent set consisting of m vertices. In this paper, we determine the Ramsey number with respect to some classes of partial order graphs. Furthermore, some implications of Ramsey numbers in ring theory are discussed. |
| first_indexed | 2024-12-14T12:31:27Z |
| format | Article |
| id | doaj.art-14e6713033d44bb5981a68753470c84f |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-12-14T12:31:27Z |
| publishDate | 2020-12-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-14e6713033d44bb5981a68753470c84f2022-12-21T23:01:09ZengDe GruyterOpen Mathematics2391-54552020-12-011811645165710.1515/math-2020-0085math-2020-0085Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theoryBadawi Ayman0Rissner Roswitha1Department of Mathematics and Statistics, College of Arts and Sciences, American University of Sharjah, Sharjah, United Arab EmiratesInstitut für Mathematik, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt am Wörthersee, AustriaFor a partially ordered set (A,≤)(A,\le ), let GA{G}_{A} be the simple, undirected graph with vertex set A such that two vertices a≠b∈Aa\ne b\in A are adjacent if either a≤ba\le b or b≤ab\le a. We call GA{G}_{A} the partial order graph or comparability graph of A. Furthermore, we say that a graph G is a partial order graph if there exists a partially ordered set A such that G=GAG={G}_{A}. For a class C{\mathcal{C}} of simple, undirected graphs and n, m≥1m\ge 1, we define the Ramsey number ℛC(n,m){ {\mathcal R} }_{{\mathcal{C}}}(n,m) with respect to C{\mathcal{C}} to be the minimal number of vertices r such that every induced subgraph of an arbitrary graph in C{\mathcal{C}} consisting of r vertices contains either a complete n-clique Kn{K}_{n} or an independent set consisting of m vertices. In this paper, we determine the Ramsey number with respect to some classes of partial order graphs. Furthermore, some implications of Ramsey numbers in ring theory are discussed.http://www.degruyter.com/view/j/math.2020.18.issue-1/math-2020-0085/math-2020-0085.xml?format=INTramsey numberpartial orderpartial order graphinclusion graph06a0605cxx05d10 |
| spellingShingle | Badawi Ayman Rissner Roswitha Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory Open Mathematics ramsey number partial order partial order graph inclusion graph 06a06 05cxx 05d10 |
| title | Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory |
| title_full | Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory |
| title_fullStr | Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory |
| title_full_unstemmed | Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory |
| title_short | Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory |
| title_sort | ramsey numbers of partial order graphs comparability graphs and implications in ring theory |
| topic | ramsey number partial order partial order graph inclusion graph 06a06 05cxx 05d10 |
| url | http://www.degruyter.com/view/j/math.2020.18.issue-1/math-2020-0085/math-2020-0085.xml?format=INT |
| work_keys_str_mv | AT badawiayman ramseynumbersofpartialordergraphscomparabilitygraphsandimplicationsinringtheory AT rissnerroswitha ramseynumbersofpartialordergraphscomparabilitygraphsandimplicationsinringtheory |