On-line Ramsey Theory

Grytczuk, Jarosław; Hałuszczak, Mariusz; Kierstead, H. A.

doi:10.37236/1810

Cited by 28 publications

(55 citation statements)

References 3 publications

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“…. For a fixed graph H, studying the ratio of r H ( ) and R H ( ) e was initiated in [2,10,14] and has drawn much attention since then [11][12][13]20]. There is also a line of research trying to determine some exact online Ramsey numbers [5,7,8,12,20,21].…”

Section: Introductionmentioning

confidence: 99%

“…She showed that one direction of the conjecture is true while the other direction is not. Theorem 1.3 (Grytczuk et al [11]). Painter wins the online Ramsey game for C 3 on outerplanar graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Painter wins the online Ramsey game for C 3 on outerplanar graphs. Proposition 1.4 (Grytczuk et al [11]). Builder wins the online Ramsey game for C 3 on 2-degenerate planar graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Grytczuk, Hałuszczak, and Kierstead [11] proved that Builder wins the online Ramsey game for C 3 on 2-degenerate planar graphs, but Painter wins the online Ramsey game for C 3 on…”

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confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Painter wins the online Ramsey game for C 3 on outerplanar graphs. Proposition 1.4 (Grytczuk et al [11]). Builder wins the online Ramsey game for C 3 on 2-degenerate planar graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Grytczuk, Hałuszczak, and Kierstead [11] proved that Builder wins the online Ramsey game for C 3 on 2-degenerate planar graphs, but Painter wins the online Ramsey game for C 3 on…”

mentioning

confidence: 99%

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Online Ramsey theory for a triangle on ‐free graphs

Choi¹,

Choi

Jeong³

et al. 2019

Journal of Graph Theory

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“…Recall the statement of Theorem 5.Theorem 5. If H is a 3-uniform hypergraph, then τ g (H) ≤ 516 (n H + m H ). As a further consequence of Theorem 8, we have the following upper bound on the game transversal number of a 3-uniform hypergraph with maximum degree at most 2.…”

mentioning

confidence: 99%

Bounds on the game transversal number in hypergraphs

Bujts

Henning

Tuza

2017

European Journal of Combinatorics

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Let H = (V, E) be a hypergraph with vertex set V and edge set E of order n H = |V | and size m H = |E|. A transversal in H is a subset of vertices in H that has a nonempty intersection with every edge of H. A vertex hits an edge if it belongs to that edge. The transversal game played on H involves of two players, Edge-hitter and Staller, who take turns choosing a vertex from H. Each vertex chosen must hit at least one edge not hit by the vertices previously chosen. The game ends when the set of vertices chosen becomes a transversal in H. Edge-hitter wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The game transversal number, τ g (H), of H is the number of vertices chosen when Edge-hitter starts the game and both players play optimally. We compare the game transversal number of a hypergraph with its transversal number, and also present an important fact concerning the monotonicity of τ g , that we call the Transversal Continuation Principle. It is known that if H is a hypergraph with all edges of size at least 2, and H is not a 4-cycle, then τ g (H) ≤ 4 11 (n H + m H ); and if H is a (loopless) graph, then τ g (H) ≤ 1 3 (n H + m H + 1). We prove that if H is a 3-uniform hypergraph, then τ g (H) ≤ 5 16 (n H + m H ), and if H is 4-uniform, then τ g (H) ≤ 71 252 (n H + m H ).

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Coloring random graphs online without creating monochromatic subgraphs

Mütze

Rast

Spöhel

2012

Random Struct Algorithms

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Consider the following random process: The vertices of a binomial random graph Gn,p are revealed one by one, and at each step only the edges induced by the already revealed vertices are visible. Our goal is to assign to each vertex one from a fixed number r of available colors immediately and irrevocably without creating a monochromatic copy of some fixed graph F in the process. Our first main result is that for any F and r, the threshold function for this problem is given by p0(F,r,n) = n‐1/m*1(F,r), where m*1(F,r) denotes the so‐called online vertex‐Ramsey density of F and r. This parameter is defined via a purely deterministic two‐player game, in which the random process is replaced by an adversary that is subject to certain restrictions inherited from the random setting. Our second main result states that for any F and r, the online vertex‐Ramsey density m*1(F,r) is a computable rational number. Our lower bound proof is algorithmic, i.e., we obtain polynomial‐time online algorithms that succeed in coloring Gn,p as desired with probability 1 ‐ o(1) for any p(n) = o(n‐1/m*1(F,r)). © 2012 Wiley Periodicals, Inc. Random Struct. Alg. 44, 419–464, 2014

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On-line Ramsey Theory

Cited by 28 publications

References 3 publications

Online Ramsey theory for a triangle on ‐free graphs

Online Ramsey theory for a triangle on ‐free graphs

Bounds on the game transversal number in hypergraphs

Coloring random graphs online without creating monochromatic subgraphs

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