A note on on-line Ramsey numbers for quadrilaterals

• Autorzy: Cyman J. , Dzido T.

Identyfikatory

DOI

http://dx.doi.org/10.7494/OpMath.2014.34.3.463

• Języki publikacji: EN

Abstrakty

EN

We consider on-line Ramsey numbers defined by a game played between two players, Builder and Painter. In each round Builder draws an the edge and Painter colors it either red or blue, as it appears. Builder’s goal 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 (H) of the graph H. An asymmetric version of the on-line Ramsey numbers r(G,H) is defined accordingly. In 2005, Kurek and Ruciński computed r(C3). In this paper, we compute r(C4,Ck) for 3 ≤k ≤ 7. Most of the results are based on computer algorithms but we obtain the exact value r(C4) and do so without the help of computer algorithms.

Słowa kluczowe

EN

Ramsey theory; on-line games

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2014
• Tom: Vol. 34, no. 3
• Strony: 463--468
• Opis fizyczny: Bibliogr. 6 poz., rys., tab.

Twórcy

autor

Cyman J.

• Gdańsk University of Technology Department of Technical Physics and Applied Mathematics Narutowicza 11/12, 80-952 Gdańsk, Poland

autor

Dzido T.

• University of Gdańsk Institute of Informatics Wita Stwosza 57, 80-952 Gdańsk, Poland

Bibliografia

• [1] J. Beck, Achievement games and the probabilistic method [in:] Combinatorics, Paul Erdos is Eighty, Bolyai Soc. Math. Stud. 1 (1993), 51–78.
• [2] J. Cyman, T. Dzido, J. Lapinskas, A. Lo, On-line Ramsey numbers of paths and cycles, submitted.
• [3] A. Kurek, A. Ruciński, Two variants of the size Ramsey number, Discussiones Mathematicae Graph Theory 25 (2005), 141–149.
• [4] J.A. Grytczuk, H.A. Kierstead, P. Prałat, On-line Ramsey numbers for paths and stars, Discrete Mathematics and Theoretical Computer Science 10 (2008), 63–74.
• [5] P. Prałat, er(3, 4) = 17, Electronic Journal of Combinatorics 15 (2008) #R67.
• [6] S.P. Radziszowski, Small Ramsey Numbers, Electronic Journal of Combinatorics, Dynamic Survey 1, http://www.combinatorics.org.