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The Ramsey number r(C₇,C₇,C₇)

100%

Faudree R., Schelten A., Schiermeyer I.

Discussiones Mathematicae Graph Theory

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2003

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tom 23

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nr 1

141-158

EN

Bondy and Erdős [2] have conjectured that the Ramsey number for three cycles Cₖ of odd length has value r(Cₖ,Cₖ,Cₖ) = 4k-3. We give a proof that r(C₇,C₇,C₇) = 25 without using any computer support.

2

One More Turán Number and Ramsey Number for the Loose 3-Uniform Path of Length Three

100%

Polcyn J.

Discussiones Mathematicae Graph Theory

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2017

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tom 37

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nr 2

443-464

EN

Let P denote a 3-uniform hypergraph consisting of 7 vertices a, b, c, d, e, f, g and 3 edges {a, b, c}, {c, d, e}, and {e, f, g}. It is known that the r-color Ramsey number for P is R(P; r) = r + 6 for r ≤ 9. The proof of this result relies on a careful analysis of the Turán numbers for P. In this paper, we refine this analysis further and compute the fifth order Turán number for P, for all n. Using this number for n = 16, we confirm the formula R(P; 10) = 16.

3

The cycle-complete graph Ramsey number r(C₅,K₇)

100%

Schiermeyer I.

Discussiones Mathematicae Graph Theory

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2005

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tom 25

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nr 1-2

129-139

EN

The cycle-complete graph Ramsey number r(Cₘ,Kₙ) is the smallest integer N such that every graph G of order N contains a cycle Cₘ on m vertices or has independence number α(G) ≥ n. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that r(Cₘ,Kₙ) = (m-1)(n-1)+1 for all m ≥ n ≥ 3 (except r(C₃,K₃) = 6). This conjecture holds for 3 ≤ n ≤ 6. In this paper we will present a proof for r(C₅,K₇) = 25.

4

On 1-dependent ramsey numbers for graphs

80%

Cockayne E., Mynhardt C.

Discussiones Mathematicae Graph Theory

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1999

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tom 19

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nr 1

93-110

EN

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.

Ograniczanie wyników

4 Discussiones Mathematicae Graph Theory

2 Schiermeyer I.

1 Cockayne E.

1 Faudree R.

1 Mynhardt C.

1 Polcyn J.

1 Schelten A.

1 2017

1 2005

1 2003

1 1999

The Ramsey number r(C₇,C₇,C₇)

One More Turán Number and Ramsey Number for the Loose 3-Uniform Path of Length Three

The cycle-complete graph Ramsey number r(C₅,K₇)

On 1-dependent ramsey numbers for graphs