On a Ramsey Problem Involving Quadrilaterals

Jayawardene, Chula J.; Hewage, T.U.

doi:10.22457/apam.v13n2a16

Cited by 2 publications

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“…Given any two graphs G and H, the classical Ramsey number (see [2,4,7,8]) r(H,G) is defined as the smallest positive integer n such that K n → (H,G). A natural generalization of the popular classical Ramsey number is the size multipartite Ramsey number which was introduced a few decades ago (see [1,9]).…”

Section: Introductionmentioning

confidence: 99%

A Size Multipartite Ramsey Problem Involving the Claw Graph

Jayawardene¹

2018

APAM

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Let K j×s denote a complete balanced multipartite graph consisting of j partite sets of uniform size s. For any two colouring of the edges of a graph K j×s , we say that K j×s → (K 1,3 ,G), if there exists a copy of K 1,3 (Claw graph) in the first colour or a copy of G in the second colour. m j (K 1,3 ,G) is defined as the smallest positive integer s such that K j×s → (K 1,3 ,G). In this paper we find all such m j (K 1,3 ,G) for all graphs G on 4 vertices.

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Section: Introductionmentioning

confidence: 99%

A Size Multipartite Ramsey Problem Involving the Claw Graph

Jayawardene¹

2018

APAM

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Tripartite and Quadpartite Size Ramsey Numbers for All Pairs of Connected Graphs on Four Vertices

Jayawardene

2020

Handbook of Research on Advanced Applications of Graph Theory in Modern Society

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A popular area of graph theory is based on a paper written in 1930 by F. P. Ramsey titled “On a Problem on Formal Logic.” A theorem which was proved in his paper triggered the study of modern Ramsey theory. However, his premature death at the young age of 26 hindered the development of this area of study at the initial stages. The balanced size multipartite Ramsey number mj (H,G) is defined as the smallest positive number s such that Kj×s→ (H,G). There are 36 pairs of (H, G), when H, G represent connected graphs on four vertices (as there are only 6 non-isomorphic connected graphs on four vertices). In this chapter, the authors find mj (H, G) exhaustively for all such pairs in the tripartite case j=3, and in the quadpartite case j=4, excluding the case m4 (K4,K4). In this case, the only known result is that m4 (K4,K4) is greater than or equal to 4, since no upper bound has been found as yet.

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On a Ramsey Problem Involving Quadrilaterals

Abstract: Abstract. Let j ≥ 3. Given any two coloring (consisting of say red and blue colors) of the edges of a complete graph K j×s , we say that K j×s →

Cited by 2 publications

References 5 publications

A Size Multipartite Ramsey Problem Involving the Claw Graph

A Size Multipartite Ramsey Problem Involving the Claw Graph

Tripartite and Quadpartite Size Ramsey Numbers for All Pairs of Connected Graphs on Four Vertices

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