Three color Ramsey numbers for graphs with at most 4 vertices
type: Item
instance of: scholarly article
title: Three color Ramsey numbers for graphs with at most 4 vertices
zbMATH Open document ID: 1266.05085
author: Luis Boza
author: Janusz Dybizbański
author: Tomasz Dzido
published in: The Electronic Journal of Combinatorics
publication date: 2013-06-07T00:00:00Z
full work available at URL: v19i4p47
review text: Summary: For given graphs \(H_{1}, H_{2}, H_{3}\), the 3-color Ramsey number \(R(H_{1},H_{2}, H_{3})\) is the smallest integer \(n\) such that if we arbitrarily color the edges of the complete graph of order \(n\) with 3 colors, then it always contains a monochromatic copy of \(H_{i}\) colored with \(i\), for some \(1 \leq i \leq 3\). We study the bounds on 3-color Ramsey numbers \(R(H_1,H_2,H_3)\), where \(H_i\) is an isolate-free graph different from \(K_2\) with at most four vertices, establishing that \(R(P_4,C_4,K_4) = 14\), \(R(C_4,K_3,K_4-e) = 17\), \(R(C_4,K_3+e,K_4-e) = 17\), \(R(C_4,K_4-e,K_4-e) = 19\), \(28\leq R(C_4,K_4-e,K_4) \leq 36\), \(R(K_3,K_4-e,K_4) \leq 41\), \(R(K_4-e,K_4-e,K_4) \leq 59\) and \(R(K_4-e,K_4,K_4) \leq 113\). Also, we prove that \(R(K_3+e,K_4-e,K_4-e) = R(K_3,K_4-e,K_4-e)\), \(R(C_4,K_3+e,K_4) \leq \max\{R(C_4,K_3,K_4),29\} \leq 32\), \(R(K_3+e,K_4-e,K_4) \leq \max\{R(K_3,K_4-e,K_4),33\} \leq 41\) and \(R(K_3+e,K_4,K_4)\leq \max\{R(K_3,K_4,K_4),2R(K_3,K_3,K_4)+2\} \leq79\). This paper is an extension of the article by \textit{J. Arste} et al. [Util. Math. 49, 85--96 (1996; Zbl 0854.05077)].
Mathematics Subject Classification ID: 05C55
Mathematics Subject Classification ID: 05D10
Mathematics Subject Classification ID: 05C35
zbMATH DE Number: 6171830
zbMATH Keywords: extremal graphs
zbMATH Keywords: Ramsey numbers
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MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6171830