The Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths

Halina Bielak, Kinga Dąbrowska

Abstract


The Ramsey number \(R(G, H)\) for a pair of graphs \(G\) and \(H\) is defined as the smallest integer \(n\) such that, for any graph \(F\) on \(n\) vertices, either \(F\) contains \(G\) or \(\overline{F}\) contains \(H\) as a subgraph, where \(\overline{F}\) denotes the complement of \(F\). We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers \(R(K_1+L_n, P_m)\) and \(R(K_1+L_n, C_m)\) for some integers \(m\), \(n\), where \(L_n\) is a linear forest of order \(n\) with at least one edge.

Keywords


Cycle; path; Ramsey number; Turan number

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DOI: http://dx.doi.org/10.17951/a.2015.69.2.1-7
Data publikacji: 2015-12-30 22:51:59
Data złożenia artykułu: 2015-12-29 21:01:57

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Copyright (c) 2015 Halina Bielak, Kinga Dąbrowska