On-line Ramsey numbers for paths and stars

J. A. Grytczuk, Henry Kierstead, P. Prałlat

Research output: Contribution to journalArticle

18 Scopus citations

Abstract

We study on-line version of size-Ramsey numbers of graphs defined via a game played between Builder and Painter: in one round Builder joins two vertices by an edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible. The minimum,number of rounds (assuming both players play perfectly) is the on-line Ramsey number r̃(H) of the graph H. We determine exact values of r̃(H) for a few short paths and obtain a general upper bound r̃(Pn) ≤ 4n - 7. We also study asymmetric version of this parameter when one of the target graphs is a star Sn with n edges. We prove that r̃(Sn, H) ≤ n · e(H) when H is any tree, cycle or clique.

Original languageEnglish (US)
Pages (from-to)63-74
Number of pages12
JournalDiscrete Mathematics and Theoretical Computer Science
Volume10
Issue number3
StatePublished - Oct 13 2008

Keywords

  • Online Ramsey games
  • Size Ramsey number

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)
  • Discrete Mathematics and Combinatorics

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