A note on on-line Ramsey numbers for quadrilaterals

Abstract

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 $\widetilde{r}(H)$ of the graph $H$. An asymmetric version of the on-line Ramsey numbers $\widetilde{r}(G,H)$ is defined accordingly. In 2005, Kurek and Ruciński computed $\widetilde{r}(C_3)$. In this paper, we compute $\widetilde{r}(C_4,C_k)$ for $3 \le k \le 7$. Most of the results are based on computer algorithms but we obtain the exact value $\widetilde{r}(C_4)$ and do so without the help of computer algorithms.