The random planar graph process

Gerke, Stefanie; Schlatter, Dirk; Steger, Angelika; Taraz, Anusch

doi:10.1002/rsa.20186

Cited by 24 publications

(20 citation statements)

References 13 publications

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“…Lemmas 4.3 and 4.4 (for g ≥ 1) and [43,Lemma 2] (for g = 0) tell us that the cubic kernel K g (2l, 3l) has O p (1) components. Thus by (38), we have k = O p (1) if the kernel is cubic, which is the case whp in 1Sup by Theorem 5.1. In Int, we have d(G) = O p (1).…”

Section: 2mentioning

confidence: 78%

Phase transitions in graphs on orientable surfaces

Kang

Moßhammer

Sprüssel

2020

Random Struct Algorithms

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Let S g be the orientable surface of genus g and denote by  g (n, m) the class of all graphs on vertex set [n] = {1, … , n} with m edges embeddable on S g . We prove that the component structure of a graph chosen uniformly at random from  g (n, m) features two phase transitions. The first phase transition mirrors the classical phase transition in the Erdős-Rényi random graph G(n, m) chosen uniformly at random from all graphs with vertex set [n] and m edges. It takes place at m = n 2 + O(n 2∕3 ), when the giant component emerges. The second phase transition occurs at m = n + O(n 3∕5 ), when the giant component covers almost all vertices of the graph. This kind of phenomenon is strikingly different from G(n, m) and has only been observed for graphs on surfaces.

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Section: 2mentioning

confidence: 78%

Phase transitions in graphs on orientable surfaces

Kang

Moßhammer

Sprüssel

2020

Random Struct Algorithms

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“…This study was pioneered by Ruciński and Wormald in 1992 [25] for graphs with a fixed degree sequence, and also by Erdős, Suen, and Winkler in 1995 for triangle-free and bipartite graphs [11]. Since then, many other graph properties have been studied, such as planarity [17], H-freeness [23], and also the property of being intersecting in the context of hypergraphs [6]. Thus our model is a natural extension of this approach to the satisfiability setting.…”

Section: Our Contributionmentioning

confidence: 99%

On the Random Satisfiable Process

Krivelevich¹,

Sudakov²,

Vilenchik³

2009

Combinator. Probab. Comp.

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Abstract. In this work we suggest a new model for generating random satisfiable k-CNF formulas. To generate such formulas -randomly permute all 2 k`n k´p ossible clauses over the variables x1, . . . , xn, and starting from the empty formula, go over the clauses one by one, including each new clause as you go along if after its addition the formula remains satisfiable. We study the evolution of this process, namely the distribution over formulas obtained after scanning through the first m clauses (in the random permutation's order). Random processes with conditioning on a certain property being respected are widely studied in the context of graph properties. This study was pioneered by Ruciński and Wormald in 1992 for graphs with a fixed degree sequence, and also by Erdős, Suen, and Winkler in 1995 for triangle-free and bipartite graphs. Since then many other graph properties were studied such as planarity and H-freeness. Thus our model is a natural extension of this approach to the satisfiability setting. Our main contribution is as follows. For m ≥ cn, c = c(k) a sufficiently large constant, we are able to characterize the structure of the solution space of a typical formula in this distribution. Specifically, we show that typically all satisfying assignments are essentially clustered in one cluster, and all but e −Ω(m/n) n of the variables take the same value in all satisfying assignments. We also describe a polynomial time algorithm that finds whp a satisfying assignment for such formulas.

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“…Monotonicity of

𝒫

guarantees that the final graph G N is maximal in

𝒫

and facilitates the use of some common techniques such as coupling with a modified process. So far, global properties are far less well understood and there is no standard approach to analyzing these processes (see for instance, the properties of being planar 19, r ‐colourable 26 and k ‐matching‐free 23).…”

Section: Introductionmentioning

confidence: 99%

The Kőnig graph process

Kamčev

Krivelevich

Morrison

et al. 2020

Random Struct Algorithms

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Say that a graph G has property  if the size of its maximum matching is equal to the order of a minimal vertex cover. We study the following process. Set N ∶= (n 2) and let e 1 , e 2 , … e N be a uniformly random ordering of the edges of K n , with n an even integer. Let G 0 be the empty graph on n vertices. For m ≥ 0, G m+1 is obtained from G m by adding the edge e m+1 exactly if G m ∪{e m+1 } has property . We analyze the behavior of this process, focusing mainly on two questions: What can be said about the structure of G N and for which m will G m contain a perfect matching? KEYWORDS matching, perfect matching, random graph, random process, vertex cover 1 INTRODUCTION The modern study of random graph processes began in 1959 with the inaugural papers of Erdős and Rényi [10, 11]. Given a uniformly random permutation e 1 , … , e N of E(K n), they studied the evolution and properties of the graph G n,m with edge set {e 1 , … , e m }, which is now known as the Erdős-Rényi random graph. This work has since grown into a well-established research area with many important applications in theoretical computer science, statistical physics, and other branches of mathematics [5, 17, 20]. An important variant of the standard Erdős-Rényi process, often referred to as the random greedy process, is the following. Given a graph property , preserved by the removal of edges, begin with an empty n-vertex graph and at each step add an edge chosen uniformly at random from those that do not violate property . The random greedy process was first considered by Ruciński and Wormald [27] (in the case of bounded degree) and, following discussions of Bollobás and Erdős, by Erdős, Suen and Winkler in 1995 [13] (in the case of triangle-freeness). Their motivation was defining and analyzing a natural probability measure on the set of -maximal graphs.

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The random planar graph process

Cited by 24 publications

References 13 publications

Phase transitions in graphs on orientable surfaces

Phase transitions in graphs on orientable surfaces

On the Random Satisfiable Process

The Kőnig graph process

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