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On 1-dependent ramsey numbers for graphs

100%

Cockayne E., Mynhardt C.

Discussiones Mathematicae Graph Theory

1999

tom 19

nr 1

93-110

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.

An inequality chain of domination parameters for trees

81%

Cockayne E., Favaron O., Puech J., Mynhardt C.

Discussiones Mathematicae Graph Theory

1998

tom 18

nr 1

127-142

We prove that the smallest cardinality of a maximal packing in any tree is at most the cardinality of an R-annihilated set. As a corollary to this result we point out that a set of parameters of trees involving packing, perfect neighbourhood, R-annihilated, irredundant and dominating sets is totally ordered. The class of trees for which all these parameters are equal is described and we give an example of a tree in which most of them are distinct.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Cockayne E.

2 Mynhardt C.

1 Favaron O.

1 Puech J.

1 1999

1 1998

Wyniki wyszukiwania

On 1-dependent ramsey numbers for graphs

An inequality chain of domination parameters for trees