Article in press
Authors:
Title:
Online size Ramsey number for $C_4$ and $P_6$
PDFSource:
Discussiones Mathematicae Graph Theory
Received: 2023-02-15 , Revised: 2023-07-20 , Accepted: 2023-07-21 , Available online: 2023-09-10 , https://doi.org/10.7151/dmgt.2513
Abstract:
In this paper we consider a game played on the edge set of the infinite clique
$K_\mathbb{N}$ by two players, Builder and Painter. In each
round of the game, Builder chooses an edge and Painter colors
it red or blue. Builder wins when Painter creates a red copy of $G$ or a blue
copy of $H$, for some fixed graphs $G$ and $H$. Builder wants to win in as few
rounds as possible, and Painter wants to delay Builder for as many rounds as
possible.
The online size Ramsey number $\tilde{r}(G,H)$, is the minimum number
of rounds within which Builder can win, assuming both players play optimally.
So far it has been proven by Dybizbański, Dzido and Zakrzewska that
$11\leq\tilde{r}(C_4,P_6)\leq13$ [J. Dybizbański, T. Dzido and R. Zakrzewska,
On-line Ramsey numbers for paths and short cycles, Discrete Appl. Math.
282 (2020) 265–270]. In this paper, we refine this result and show the exact
value, namely we will present the theorem that $\tilde{r}(C_4,P_6)=11$, with
the details of the proof.
Keywords:
graph theory, Ramsey theory, combinatorial games, online size Ramsey number
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