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ArticleOriginal scientific text

Title

On 1-dependent ramsey numbers for graphs

Authors 1, 2

Affiliations

  1. Department of Mathematics, University of Victoria, P.O. Box 3045, Victoria, BC, CANADA V8W 3P4
  2. 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 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

ENG
1-dependence, irredundance, CO-irredundance, Ramsey numbers

Bibliography

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Discussiones Mathematicae Graph Theory
1999/19/1
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Pages:
93-110
Main language of publication
English
Received
1998-07-14
Accepted
1999-04-12
Published
1999

DOI: 10.7151/dmgt.1088

Exact and natural sciences