Online vertex-coloring games in random graphs

Marciniszyn, Martin; Spöhel, Reto

doi:10.1007/s00493-010-2436-z

Cited by 8 publications

(18 citation statements)

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“…The algorithm presented in [8] to compute m * (F, r) for general F and r is rather complex and gives no hint as to how the quantity m * (F, r) behaves for natural graph families. However, for a large class of graphs F , a simple closed formula for the parameter m * (F, r) follows from the results in [6]. This class includes cliques K , cycles C , complete bipartite graphs K s,t , d-dimensional hypercubes Q d , wheels W with spokes, and stars S with rays.…”

Section: 2mentioning

confidence: 99%

“…For the rest of this paper, we restrict our attention to forests and focus on the parameter k * (F, r). It follows from the results in [6] that for any tree F and any integer r ≥ 2 the greedy strategy guarantees a lower bound of k * (F, r) ≥ v(F ) r (for the sake of completeness we give the argument explicitly in Lemma 8 below). 0 7 0 2 0 16 0 0 0 27 7 5 0 41 18 32 7 55 32 Table 1.…”

Section: Corollary 3 ([8]mentioning

confidence: 99%

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On the Path-Avoidance Vertex-Coloring Game

Mütze

Spöhel

2011

Electron. J. Combin.

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For any graph F and any integer r ≥ 2, the online vertex-Ramsey density of F and r, denoted m * (F, r), is a parameter defined via a deterministic two-player Ramsey-type game (Painter vs. Builder). This parameter was introduced in a recent paper [8], where it was shown that the online vertex-Ramsey density determines the threshold of a similar probabilistic one-player game (Painter vs. the binomial random graph G n,p ). For a large class of graphs F , including cliques, cycles, complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a simple greedy strategy is optimal for Painter and closed formulas for m * (F, r) are known. In this work we show that for the case where F = P is a (long) path, the picture is very different. It is not hard to see that m * (P , r) = 1 − 1/k * (P , r) for an appropriately defined integer k * (P , r), and that the greedy strategy gives a lower bound of k * (P , r) ≥ r . We construct and analyze Painter strategies that improve on this greedy lower bound by a factor polynomial in , and we show that no superpolynomial improvement is possible.arXiv:1103.5657v1 [math.CO]

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Section: 2mentioning

confidence: 99%

Section: Corollary 3 ([8]mentioning

confidence: 99%

On the Path-Avoidance Vertex-Coloring Game

Mütze

Spöhel

2011

Electron. J. Combin.

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“…In [14], a vertex-coloring variant of the online F -avoidance game was introduced, and threshold results similar to Theorem 2 and Theorem 3 were proved for an arbitrary number of colors. Both Theorem 4 and Theorem 5 can easily be adapted to this vertex setting.…”

Section: 2mentioning

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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“…It follows from the results in [32] that for any tree F and any integer r ≥ 2 the greedy strategy (with H 1 = · · · = H r = F ) is a winning strategy for Painter in the deterministic game with tree size restriction k = v(F ) r − 1, i.e., guarantees a lower bound of k * (F, r) ≥ v(F ) r .…”

Section: 2mentioning

confidence: 99%

“…Theorem 2 ( [32]). For any fixed graph F with at least one edge and any fixed integer r ≥ 2, the threshold for finding an r-coloring of G n,p that is valid with respect to F online satisfies p 0 (F, r, n) ≥ n −1/m 1 (F,r) ,…”

Section: Introductionmentioning

confidence: 99%

Coloring random graphs online without creating monochromatic subgraphs

Mütze¹,

Rast²,

Spöhel³

2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

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Abstract. Consider the following random process: The vertices of a binomial random graph G n,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 p 0 (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 G n,p as desired with probability 1 − o(1) for any p(n) = o(n −1/m * 1 (F,r) ).

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Online vertex-coloring games in random graphs

Cited by 8 publications

References 7 publications

On the Path-Avoidance Vertex-Coloring Game

On the Path-Avoidance Vertex-Coloring Game

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

Coloring random graphs online without creating monochromatic subgraphs

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