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## Discussiones Mathematicae Graph Theory

1999 | 19 | 1 | 93-110
Tytuł artykułu

### On 1-dependent ramsey numbers for graphs

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A set X of vertices of a graph G is said to be 1-dependent if the subgraph of G induced by X has maximum degree one. The 1-dependent Ramsey number t₁(l,m) is the smallest integer n such that for any 2-edge colouring (R,B) of Kₙ, the spanning subgraph B of Kₙ has a 1-dependent set of size l or the subgraph R has a 1-dependent set of size m. The 2-edge colouring (R,B) is a t₁(l,m) Ramsey colouring of Kₙ if B (R, respectively) does not contain a 1-dependent set of size l (m, respectively); in this case R is also called a (l,m,n) Ramsey graph. We show that t₁(4,5) = 9, t₁(4,6) = 11, t₁(4,7) = 16 and t₁(4,8) = 17. We also determine all (4,4,5), (4,5,8), (4,6,10) and (4,7,15) Ramsey graphs.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
93-110
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-07-14
poprawiono
1999-04-12
Twórcy
autor
• Department of Mathematics, University of Victoria, P.O. Box 3045, Victoria, BC, CANADA V8W 3P4
autor
• Department of Mathematics, University of South Africa, P.O. Box 392, Pretoria, South Africa 0003
Bibliografia
• [1] R.C. Brewster, E.J. Cockayne and C.M. Mynhardt, Irredundant Ramsey numbers for graphs, J. Graph Theory 13 (1989) 283-290, doi: 10.1002/jgt.3190130303.
• [2] G. Chartrand and L. Lesniak, Graphs and Digraphs (Third Edition) (Chapman and Hall, London, 1996).
• [3] E.J. Cockayne, Generalized irredundance in graphs: Hereditary properties and Ramsey numbers, submitted.
• [4] E.J. Cockayne, G. MacGillivray and J. Simmons, CO-irredundant Ramsey numbers for graphs, submitted.
• [5] E.J. Cockayne, C.M. Mynhardt and J. Simmons, The CO-irredundant Ramsey number t(4,7), submitted.
• [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
• [7] H. Harborth and I Mengersen, All Ramsey numbers for five vertices and seven or eight edges, Discrete Math. 73 (1988/89) 91-98, doi: 10.1016/0012-365X(88)90136-7.
• [8] C.J. Jayawardene and C.C. Rousseau, The Ramsey numbers for a quadrilateral versus all graphs on six vertices, to appear.
• [9] G. MacGillivray, personal communication, 1998.
• [10] C.M. Mynhardt, Irredundant Ramsey numbers for graphs: a survey, Congr. Numer. 86 (1992) 65-79.
• [11] S.P. Radziszowski, Small Ramsey numbers, Electronic J. Comb. 1 (1994) DS1.
• [12] F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930) 264-286, doi: 10.1112/plms/s2-30.1.264.
• [13] J. Simmons, CO-irredundant Ramsey numbers for graphs, Master's dissertation, University of Victoria, Canada, 1998.
• [14] Zhou Huai Lu, The Ramsey number of an odd cycle with respect to a wheel, J. Math. - Wuhan 15 (1995) 119-120.
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Bibliografia
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