### 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(P_{n}) ≤ 4n - 7. We also study asymmetric version of this parameter when one of the target graphs is a star S_{n} with n edges. We prove that r̃(S_{n}, H) ≤ n · e(H) when H is any tree, cycle or clique.

Original language | English (US) |
---|---|

Pages (from-to) | 63-74 |

Number of pages | 12 |

Journal | Discrete Mathematics and Theoretical Computer Science |

Volume | 10 |

Issue number | 3 |

State | Published - 2008 |

### Fingerprint

### Keywords

- Online Ramsey games
- Size Ramsey number

### ASJC Scopus subject areas

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

### Cite this

*Discrete Mathematics and Theoretical Computer Science*,

*10*(3), 63-74.

**On-line Ramsey numbers for paths and stars.** / Grytczuk, J. A.; Kierstead, Henry; Prałlat, P.

Research output: Contribution to journal › Article

*Discrete Mathematics and Theoretical Computer Science*, vol. 10, no. 3, pp. 63-74.

}

TY - JOUR

T1 - On-line Ramsey numbers for paths and stars

AU - Grytczuk, J. A.

AU - Kierstead, Henry

AU - Prałlat, P.

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - Online Ramsey games

KW - Size Ramsey number

UR - http://www.scopus.com/inward/record.url?scp=53349178804&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=53349178804&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:53349178804

VL - 10

SP - 63

EP - 74

JO - Discrete Mathematics and Theoretical Computer Science

JF - Discrete Mathematics and Theoretical Computer Science

SN - 1365-8050

IS - 3

ER -