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Discussiones Mathematicae Graph Theory 19(1) (1999) 93-110
DOI: https://doi.org/10.7151/dmgt.1088

ON 1-DEPENDENT RAMSEY NUMBERS FOR GRAPHS

E.J. Cockayne Department of Mathematics, University of Victoria P.O. Box 3045, Victoria, BC, CANADA V8W 3P4

C.M. Mynhardt Department of Mathematics, University of South Africa P.O. Box 392, Pretoria, South Africa 0003

Abstract

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 K_n, the spanning subgraph B of K_n 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 K_n 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.

1991 Mathematics Subject Classification: 05C55, 05C70.

References

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Received 14 July 1998
Revised 12 April 1999