On-line Ramsey Theory for Bounded Degree Graphs

Butterfield, Jane; Grauman, Tracy; Kinnersley, William B.; Milans, Kevin G.; Stocker, Christopher J.; West, Douglas B.

doi:10.37236/623

Cited by 21 publications

(26 citation statements)

References 21 publications

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“…One beautiful question of this variety, due to Butterfield, Grauman, Kinnersley, Milans, Stocker and West [4], is to determine whether, given a natural number ∆, there exists d(∆) such that Builder can force Painter to draw a monochromatic copy of any graph with maximum degree ∆ by drawing only graphs of maximum degree at most d(∆).…”

Section: Resultsmentioning

confidence: 99%

On-line Ramsey Numbers

Conlon¹

2010

SIAM J. Discrete Math.

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Consider the following game between two players, Builder and Painter. Builder draws edges one at a time and Painter colours them, in either red or blue, as each appears. Builder's aim is to force Painter to draw a monochromatic copy of a fixed graph G. The minimum number of edges which Builder must draw, regardless of Painter's strategy, in order to guarantee that this happens is known as the on-line Ramsey numberr(G) of G. Our main result, relating to the conjecture that r(K t ) = o( r(t)2 ), is that there exists a constant c > 1 such thatfor infinitely many values of t. We also prove a more specific upper bound for this number, showing that there exists a constant c such thatrFinally, we prove a new upper bound for the on-line Ramsey number of the complete bipartite graph K t,t .

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Section: Resultsmentioning

confidence: 99%

On-line Ramsey Numbers

Conlon¹

2010

SIAM J. Discrete Math.

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“…Let P be such a red path with vertices x x x x , , , 4 , and x 5 in this order on P. Now, Builder draws two edges x x 2 6 and x x 4 6 with a new vertex x 6 . We claim that Painter must color both x x 2 6 and x x 4 6 with the color blue.…”

Section: F-free Graphs?mentioning

confidence: 99%

Online Ramsey theory for a triangle on ‐free graphs

Choi¹,

Choi

Jeong³

et al. 2019

Journal of Graph Theory

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Given a class scriptC of graphs and a fixed graph H, the online Ramsey game for H on scriptC is a game between two players Builder and Painter as follows: an unbounded set of vertices is given as an initial state, and on each turn Builder introduces a new edge with the constraint that the resulting graph must be in scriptC, and Painter colors the new edge either red or blue. Builder wins the game if Painter is forced to make a monochromatic copy of H at some point in the game. Otherwise, Painter can avoid creating a monochromatic copy of H forever, and we say Painter wins the game. We initiate the study of characterizing the graphs F such that for a given graph H, Painter wins the online Ramsey game for H on F‐free graphs. We characterize all graphs F such that Painter wins the online Ramsey game for C3 on the class of F‐free graphs, except when F is one particular graph. We also show that Painter wins the online Ramsey game for C3 on the class of K4‐minor‐free graphs, extending a result by Grytczuk, Hałuszczak, and Kierstead [Electron. J. Combin. 11 (2004), p. 60].

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“…In the remainder of the proof, we will argue that a blue path of length k guarantees at least 5 3 k+O(1) additional edges that are adjacent to it. This then proves our claim as we have counted at least…”

Section: Asymptotic Boundsmentioning

confidence: 99%

Probabilistic One-Player Ramsey Games via Deterministic Two-Player Games

Belfrage

Mütze

Spöhel

2012

SIAM J. Discrete Math.

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Consider the following probabilistic one-player game: The board is a graph with n vertices, which initially contains no edges. In each step, a new edge is drawn uniformly at random from all non-edges and is presented to the player, henceforth called Painter. Painter must assign one of r available colors to each edge immediately, where r ≥ 2 is a fixed integer. The game is over as soon as a monochromatic copy of some fixed graph F has been created, and Painter's goal is to 'survive' for as many steps as possible before this happens. We present a new technique for deriving upper bounds on the threshold of this game, i.e., on the typical number of steps Painter will survive with an optimal strategy. More specifically, we consider a deterministic two-player variant of the game where the edges are not chosen randomly, but by a second player Builder. However, Builder has to adhere to the restriction that, for some real number d, the ratio of edges to vertices in all subgraphs of the evolving board never exceeds d. We show that the existence of a winning strategy for Builder in this deterministic game implies an upper bound of n 2−1/d for the threshold of the original probabilistic game. Moreover, we show that the best bound that can be derived in this way is indeed the threshold of the game if F is a forest. We illustrate our technique with several examples, and derive new explicit bounds for the case when F is a path.

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On-line Ramsey Theory for Bounded Degree Graphs

Cited by 21 publications

References 21 publications

On-line Ramsey Numbers

On-line Ramsey Numbers

Online Ramsey theory for a triangle on ‐free graphs

Probabilistic One-Player Ramsey Games via Deterministic Two-Player Games

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