Geodesic
On 1-dependent ramsey numbers for graphs
Discussiones Mathematicae. Graph Theory, Tome 19 (1999) no. 1, pp. 93-110.

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: 1-dependence, irredundance, CO-irredundance, Ramsey numbers 
@article{DMGT_1999_19_1_a7,
     author = {Cockayne, E. and Mynhardt, C.},
     title = {On 1-dependent ramsey numbers for graphs},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {93--110},
     publisher = {mathdoc},
     volume = {19},
     number = {1},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_1999_19_1_a7/}
}
TY  - JOUR
AU  - Cockayne, E.
AU  - Mynhardt, C.
TI  - On 1-dependent ramsey numbers for graphs
JO  - Discussiones Mathematicae. Graph Theory
PY  - 1999
SP  - 93
EP  - 110
VL  - 19
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_1999_19_1_a7/
LA  - en
ID  - DMGT_1999_19_1_a7
ER  - 
%0 Journal Article
%A Cockayne, E.
%A Mynhardt, C.
%T On 1-dependent ramsey numbers for graphs
%J Discussiones Mathematicae. Graph Theory
%D 1999
%P 93-110
%V 19
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_1999_19_1_a7/
%G en
%F DMGT_1999_19_1_a7
Cockayne, E.; Mynhardt, C. On 1-dependent ramsey numbers for graphs. Discussiones Mathematicae. Graph Theory, Tome 19 (1999) no. 1, pp. 93-110. http://geodesic.mathdoc.fr/item/DMGT_1999_19_1_a7/

[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.