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On 1-dependent ramsey numbers for graphs

E.J. Cockayne, C.M. Mynhardt (1999)

Discussiones Mathematicae Graph Theory

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...

On a Class of Vertex Folkman Numbers

Nenov, Nedyalko (2002)

Serdica Mathematical Journal

Let a1 , . . . , ar, be positive integers, i=1 ... r, m = ∑(ai − 1) + 1 and p = max{a1 , . . . , ar }. For a graph G the symbol G → (a1 , . . . , ar ) means that in every r-coloring of the vertices of G there exists a monochromatic ai -clique of color i for some i ∈ {1, . . . , r}. In this paper we consider the vertex Folkman numbers F (a1 , . . . , ar ; m − 1) = min |V (G)| : G → (a1 , . . . , ar ) and Km−1 ⊂ G} We prove that F (a1 , . . . , ar ; m − 1) = m + 6, if p = 3 and m ≧ 6 (Theorem 3)...

On dual Ramsey theorems for relational structures

Dragan Mašulović (2020)

Czechoslovak Mathematical Journal

We discuss dual Ramsey statements for several classes of finite relational structures (such as finite linearly ordered graphs, finite linearly ordered metric spaces and finite posets with a linear extension) and conclude the paper with another rendering of the Nešetřil-Rödl Theorem for relational structures. Instead of embeddings which are crucial for ``direct'' Ramsey results, for each class of structures under consideration we propose a special class of quotient maps and prove a dual Ramsey theorem...

On Monochromatic Subgraphs of Edge-Colored Complete Graphs

Eric Andrews, Futaba Fujie, Kyle Kolasinski, Chira Lumduanhom, Adam Yusko (2014)

Discussiones Mathematicae Graph Theory

In a red-blue coloring of a nonempty graph, every edge is colored red or blue. If the resulting edge-colored graph contains a nonempty subgraph G without isolated vertices every edge of which is colored the same, then G is said to be monochromatic. For two nonempty graphs G and H without isolated vertices, the mono- chromatic Ramsey number mr(G,H) of G and H is the minimum integer n such that every red-blue coloring of Kn results in a monochromatic G or a monochromatic H. Thus, the standard Ramsey...

On path-quasar Ramsey numbers

Binlong Li, Bo Ning (2015)

Annales UMCS, Mathematica

Let G1 and G2 be two given graphs. The Ramsey number R(G1,G2) is the least integer r such that for every graph G on r vertices, either G contains a G1 or Ḡ contains a G2. Parsons gave a recursive formula to determine the values of R(Pn,K1,m), where Pn is a path on n vertices and K1,m is a star on m+1 vertices. In this note, we study the Ramsey numbers R(Pn,K1,m), where Pn is a linear forest on m vertices. We determine the exact values of R(Pn,K1∨Fm) for the cases m ≤ n and m ≥ 2n, and for the case...

On Ramsey ( K 1 , 2 , C ₄ ) -minimal graphs

Tomás Vetrík, Lyra Yulianti, Edy Tri Baskoro (2010)

Discussiones Mathematicae Graph Theory

For graphs F, G and H, we write F → (G,H) to mean that any red-blue coloring of the edges of F contains a red copy of G or a blue copy of H. The graph F is Ramsey (G,H)-minimal if F → (G,H) but F* ↛ (G,H) for any proper subgraph F* ⊂ F. We present an infinite family of Ramsey ( K 1 , 2 , C ₄ ) -minimal graphs of any diameter ≥ 4.

On subgraphs without large components

Glenn G. Chappell, John Gimbel (2017)

Mathematica Bohemica

We consider, for a positive integer k , induced subgraphs in which each component has order at most k . Such a subgraph is said to be k -divided. We show that finding large induced subgraphs with this property is NP-complete. We also consider a related graph-coloring problem: how many colors are required in a vertex coloring in which each color class induces a k -divided subgraph. We show that the problem of determining whether some given number of colors suffice is NP-complete, even for 2 -coloring...

On the Vertex Folkman Numbers Fv(2,...,2;q)

Nenov, Nedyalko (2009)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 05C55.In this paper we shall compute the Folkman numbers ... We prove also new bounds for some vertex and edge Folkman numbers.

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

Joanna Polcyn (2017)

Discussiones Mathematicae Graph Theory

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.

On-line Ramsey theory.

Grytczuk, J.A., Hałuszczak, M., Kierstead, H.A. (2004)

The Electronic Journal of Combinatorics [electronic only]

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