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 |
C.M. Mynhardt Department of Mathematics, University of South Africa
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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 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.
1991 Mathematics Subject Classification: 05C55, 05C70.
References
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Received 14 July 1998
Revised 12 April 1999
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