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Abstract
Consider a game played on the edge set of the infinite clique by two players, Builder and Painter. In each round, Builder chooses an edge and Painter colours it red or blue. Builder wins by creating either a red copy of $G$ or a blue copy of $H$ for some fixed graphs $G$ and $H$. The minimum number of rounds within which Builder can win, assuming both players play perfectly, is the \emph{on-line Ramsey number} $\tilde{r}(G,H)$. In this paper, we consider the case where $G$ is a path $P_k$. We prove that $\tilde{r}(P_3,P_{\ell+1}) = \lceil 5\ell/4\rceil = \tilde{r}(P_3,C_{\ell})$ for all $\ell \ge 5$, and determine $\tilde{r}(P_{4},P_{\ell+1})$ up to an additive constant for all $\ell \ge 3$. We also prove some general lower bounds for on-line Ramsey numbers of the form $\tilde{r}(P_{k+1},H)$.
Authors (4)
-
Tomasz Dzido
- Wydział Matematyki, Fizyki i Informatyki UG Instytut Informatyki
-
John Lapinskas
-
Allan Lo
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- Category:
- Articles
- Type:
- artykuł w czasopiśmie wyróżnionym w JCR
- Published in:
-
ELECTRONIC JOURNAL OF COMBINATORICS
no. 22,
edition 1,
pages 1 - 32,
ISSN: 1077-8926 - Language:
- English
- Publication year:
- 2015
- Bibliographic description:
- Cyman J., Dzido T., Lapinskas J., Lo A.: On-line Ramsey Numbers of Paths and Cycles// ELECTRONIC JOURNAL OF COMBINATORICS. -Vol. 22, iss. 1 (2015), s.1-32
- Sources of funding:
-
- Free publication
- Verified by:
- Gdańsk University of Technology
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