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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1088,
author = {E.J. Cockayne and C.M. Mynhardt},
title = {On 1-dependent ramsey numbers for graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {19},
year = {1999},
pages = {93-110},
zbl = {0932.05061},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1088}
}
E.J. Cockayne; C.M. Mynhardt. On 1-dependent ramsey numbers for graphs. Discussiones Mathematicae Graph Theory, Tome 19 (1999) pp. 93-110. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1088/
[000] [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. | Zbl 0686.05038
[001] [2] G. Chartrand and L. Lesniak, Graphs and Digraphs (Third Edition) (Chapman and Hall, London, 1996).
[002] [3] E.J. Cockayne, Generalized irredundance in graphs: Hereditary properties and Ramsey numbers, submitted. | Zbl 0952.05051
[003] [4] E.J. Cockayne, G. MacGillivray and J. Simmons, CO-irredundant Ramsey numbers for graphs, submitted. | Zbl 0958.05092
[004] [5] E.J. Cockayne, C.M. Mynhardt and J. Simmons, The CO-irredundant Ramsey number t(4,7), submitted.
[005] [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998). | Zbl 0890.05002
[006] [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.
[007] [8] C.J. Jayawardene and C.C. Rousseau, The Ramsey numbers for a quadrilateral versus all graphs on six vertices, to appear. | Zbl 0982.05068
[008] [9] G. MacGillivray, personal communication, 1998.
[009] [10] C.M. Mynhardt, Irredundant Ramsey numbers for graphs: a survey, Congr. Numer. 86 (1992) 65-79. | Zbl 0783.05075
[010] [11] S.P. Radziszowski, Small Ramsey numbers, Electronic J. Comb. 1 (1994) DS1.
[011] [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. | Zbl 55.0032.04
[012] [13] J. Simmons, CO-irredundant Ramsey numbers for graphs, Master's dissertation, University of Victoria, Canada, 1998.
[013] [14] Zhou Huai Lu, The Ramsey number of an odd cycle with respect to a wheel, J. Math. - Wuhan 15 (1995) 119-120.