Star versus two stripes ramsey numbers and a conjecture of schelp

A. Gyárfás, Göbor N. Sárközy

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

R. H. Schelp conjectured that if G is a graph with |V(G)| = R(P n, P n) such that δ(G) > 3|V(G)|/4, then in every 2-colouring of the edges of G there is a monochromatic P n. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree. Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching-matching Ramsey number satisfying R(S n, nK 2, nK 2) = 3n - 1. This extends R(nK 2, nK 2) = 3n - 1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp's conjecture in an asymptotic sense through a standard application of the Regularity Lemma. It is sad that we are unable to hear Dick Schelp's reaction to our work generated by his conjecture.

Original languageEnglish
Pages (from-to)179-186
Number of pages8
JournalCombinatorics Probability and Computing
Volume21
Issue number1-2
DOIs
Publication statusPublished - Jan 2012

Fingerprint

Ramsey number
Coloring
Stars
Star
Regularity Lemma
Matching number
Imply
Path
Minimum Degree
Graph in graph theory
Colouring

ASJC Scopus subject areas

  • Applied Mathematics
  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Statistics and Probability

Cite this

Star versus two stripes ramsey numbers and a conjecture of schelp. / Gyárfás, A.; Sárközy, Göbor N.

In: Combinatorics Probability and Computing, Vol. 21, No. 1-2, 01.2012, p. 179-186.

Research output: Contribution to journalArticle

@article{3f19f463d4c7493ab2bfe57d1cf3315b,
title = "Star versus two stripes ramsey numbers and a conjecture of schelp",
abstract = "R. H. Schelp conjectured that if G is a graph with |V(G)| = R(P n, P n) such that δ(G) > 3|V(G)|/4, then in every 2-colouring of the edges of G there is a monochromatic P n. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree. Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching-matching Ramsey number satisfying R(S n, nK 2, nK 2) = 3n - 1. This extends R(nK 2, nK 2) = 3n - 1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp's conjecture in an asymptotic sense through a standard application of the Regularity Lemma. It is sad that we are unable to hear Dick Schelp's reaction to our work generated by his conjecture.",
author = "A. Gy{\'a}rf{\'a}s and S{\'a}rk{\"o}zy, {G{\"o}bor N.}",
year = "2012",
month = "1",
doi = "10.1017/S0963548311000599",
language = "English",
volume = "21",
pages = "179--186",
journal = "Combinatorics Probability and Computing",
issn = "0963-5483",
publisher = "Cambridge University Press",
number = "1-2",

}

TY - JOUR

T1 - Star versus two stripes ramsey numbers and a conjecture of schelp

AU - Gyárfás, A.

AU - Sárközy, Göbor N.

PY - 2012/1

Y1 - 2012/1

N2 - R. H. Schelp conjectured that if G is a graph with |V(G)| = R(P n, P n) such that δ(G) > 3|V(G)|/4, then in every 2-colouring of the edges of G there is a monochromatic P n. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree. Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching-matching Ramsey number satisfying R(S n, nK 2, nK 2) = 3n - 1. This extends R(nK 2, nK 2) = 3n - 1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp's conjecture in an asymptotic sense through a standard application of the Regularity Lemma. It is sad that we are unable to hear Dick Schelp's reaction to our work generated by his conjecture.

AB - R. H. Schelp conjectured that if G is a graph with |V(G)| = R(P n, P n) such that δ(G) > 3|V(G)|/4, then in every 2-colouring of the edges of G there is a monochromatic P n. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree. Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching-matching Ramsey number satisfying R(S n, nK 2, nK 2) = 3n - 1. This extends R(nK 2, nK 2) = 3n - 1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp's conjecture in an asymptotic sense through a standard application of the Regularity Lemma. It is sad that we are unable to hear Dick Schelp's reaction to our work generated by his conjecture.

UR - http://www.scopus.com/inward/record.url?scp=84859329932&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859329932&partnerID=8YFLogxK

U2 - 10.1017/S0963548311000599

DO - 10.1017/S0963548311000599

M3 - Article

AN - SCOPUS:84859329932

VL - 21

SP - 179

EP - 186

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 1-2

ER -