Sets that are connected in two random graphs

Molloy, Michael

doi:10.1002/rsa.20491

Cited by 4 publications

(12 citation statements)

References 14 publications

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“…In [44], we defined a very natural, and much simpler problem of this nature, and analyzed its core. Our primary motivation was to develop a technique to apply to the analysis of the Kempe core.…”

Section: Kempe-stripmentioning

confidence: 99%

“…We adapt the argument from [44], where we analyzed a very similar core, but in a simpler setting. In fact, the main motivation for [44] was to develop a technique that we could use here to analyze the Kempe core. The reader might prefer to read [44] before reading this section.…”

Section: The Kempe Core Thresholdmentioning

confidence: 99%

“…In fact, the main motivation for [44] was to develop a technique that we could use here to analyze the Kempe core. The reader might prefer to read [44] before reading this section. We let G n1,n2,p denote the random bipartite graph whose parts have size n 1 , n 2 and where each of the n 1 n 2 possible edges is present independently with probability p. We will need the following concentration bound on the size of the giant component of G n1,n2,p .…”

Section: The Kempe Core Thresholdmentioning

confidence: 99%

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The Freezing Threshold for k -Colourings of a Random Graph

Molloy

2018

J. ACM

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We determine the exact value of the freezing threshold, r f k , for k -colourings of a random graph when k ≥ 14. We prove that for random graphs with density above r f k , almost every colouring is such that a linear number of vertices are frozen, meaning that their colour cannot be changed by a sequence of alterations whereby we change the colours of o ( n ) vertices at a time, always obtaining another proper colouring. When the density is below r f k , then almost every colouring is such that every vertex can be changed by a sequence of alterations where we change O (log n ) vertices at a time. Frozen vertices are a key part of the clustering phenomena discovered using methods from statistical physics. The value of the freezing threshold was previously determined by the nonrigorous cavity method.

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Section: Kempe-stripmentioning

confidence: 99%

Section: The Kempe Core Thresholdmentioning

confidence: 99%

Section: The Kempe Core Thresholdmentioning

confidence: 99%

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The Freezing Threshold for k -Colourings of a Random Graph

Molloy

2018

J. ACM

Self Cite

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show abstract

Section: The Kempe Core Thresholdmentioning

confidence: 99%

“…In fact, the main motivation for [44] was to develop a technique that we could use here to analyze the Kempe core. The reader might prefer to read [44] before reading this section.…”

Section: The Kempe Core Thresholdmentioning

confidence: 99%

The freezing threshold for k-colourings of a random graph

Molloy

2012

Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing

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We rigorously determine the exact freezing threshold, r f k , for k-colourings of a random graph. We prove that for random graphs with density above r f k , almost every colouring is such that a linear number of variables are frozen, meaning that their colours cannot be changed by a sequence of alterations whereby we change the colours of o(n) vertices at a time, always obtaining another proper colouring. When the density is below r f k , then almost every colouring has at most o(n) frozen variables. This confirms hypotheses made using the non-rigorous cavity method.It has been hypothesized that the freezing threshold is the cause of the "algorithmic barrier", the long observed phenomenon that once the edge-density of a random graph passes 1 2 k ln k(1 + o k (1)), no algorithms are proven to find k-colourings, despite the fact that this density is only half the k-colourability threshold.We also show that r f k is the threshold of a strong form of reconstruction for k-colourings of the Galton-Watson tree, and of the graphical model. *

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The Size of the Giant Joint Component in a Binomial Random Double Graph

Jerrum

Makai

2021

Electron. J. Combin.

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We study the joint components in a random 'double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices. A joint component is a maximal set of vertices that supports both a red and a blue spanning tree. We show that there are critical pairs of red and blue edge densities at which a giant joint component appears. In contrast to the standard binomial graph model, the phase transition is first order: the size of the largest joint component jumps from $O(1)$ vertices to $\Theta(n)$ at the critical point. We connect this phenomenon to the properties of a certain bicoloured branching process.

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Sets that are connected in two random graphs

Cited by 4 publications

References 14 publications

The Freezing Threshold for k -Colourings of a Random Graph

The Freezing Threshold for k -Colourings of a Random Graph

The freezing threshold for k-colourings of a random graph

The Size of the Giant Joint Component in a Binomial Random Double Graph

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