Sudden Emergence of a Giantk-Core in a Random Graph

Pittel, Boris; Spencer, Joel; Wormald, Nicholas C.

doi:10.1006/jctb.1996.0036

Cited by 412 publications

(504 citation statements)

References 25 publications

(20 reference statements)

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“…If J ≥ J then |A∩A | is bounded by (14), and if J ≤ J then |A∩A | is bounded by (15). Therefore, w.h.p.…”

Section: Proofmentioning

confidence: 95%

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Sets that are connected in two random graphs

Molloy

2013

Random Struct Algorithms

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“…If J ≥ J then |A∩A | is bounded by (14), and if J ≤ J then |A∩A | is bounded by (15). Therefore, w.h.p.…”

Section: Proofmentioning

confidence: 95%

“…The proof of our theorem will be reminiscent of studies of the k-core, the pure literal rule, and other similar problems [14,11,6,7,9]. There, one repeatedly removes vertices (literals, etc.)…”

Section: Introductionmentioning

confidence: 99%

Sets that are connected in two random graphs

Molloy

2013

Random Struct Algorithms

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“…The sharpness of the threshold for the emergence of a non trivial k-core has already been proved in [7], where the threshold function is computed precisely for every k ≥ 3. It is shown that the threshold function is λ k n for a constant λ k > 0 which is computed in the paper.…”

Section: Letmentioning

confidence: 93%

The property of having a k ‐regular subgraph has a sharp threshold

Letzter

2012

Random Struct Algorithms

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“…The first determination of the threshold of existence of a k-core in a random graph was given by Pittel, Spencer and the second author [18].…”

Section: Preliminariesmentioning

confidence: 99%

Cores of random r-partite hypergraphs

Botelho

Wormald

Ziviani

2012

Information Processing Letters

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We show that the threshold c r,k for appearance of a k-core in a random r-partite r-uniform hypergraph G r,n,m is the same as for a random r-uniform hypergraph with cn/r edges without the r-partite restriction, where r, k ≥ 2. In both cases, the average degree is c. This is an important problem in the analysis of the algorithm presented in [2]. The algorithm constructs a family of minimal perfect hash functions based on random r-partite r-uniform hypergraphs with an empty k-core subgraph, for k ≥ 2. The above claim was not proved but was provided with strong experimental evidence. For an input key set S with m keys, the algorithm was the first one capable of constructing a simple and efficient family of minimal perfect hash functions that can be stored in O(m) bits, where the hidden constant is within a factor of two from the information theoretical lower bound. The case r, k = 2 was analyzed in [3] but the general case r ≥ 3, k ≥ 2 was still open.

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Sudden Emergence of a Giantk-Core in a Random Graph

Cited by 412 publications

References 25 publications

Sets that are connected in two random graphs

Sets that are connected in two random graphs

The property of having a k ‐regular subgraph has a sharp threshold

Cores of random r-partite hypergraphs

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