Cycle Factors and Renewal Theory

Kahn, Jeffry; Lubetzky, Eyal; Wormald, Nicholas C.

doi:10.1002/cpa.21613

Cited by 6 publications

(7 citation statements)

References 29 publications

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“…Though we haven't much non‐verbal evidence, this suggestion does seem to have received quite a bit of attention, but, absent any serious progress, seems not to have produced anything in print. Here and in the companion paper we establish Conjecture for combs. Theorem There exists some fixed

C

such that for every

n

and

k | n

, the random graph

G (n, C \frac{\log n}{n})

w.h.p.…”

Section: Introductionmentioning

confidence: 92%

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The threshold for combs in random graphs

Lubetzky

Wormald

2015

Random Struct. Alg.

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For k | n let Comb n,k denote the tree consisting of an (n/k)-vertex path with disjoint k-vertex paths beginning at each of its vertices. An old conjecture says that for any k = k(n) the threshold for the random graph G(n, p) to contain Comb n,k is at p log n n . Here we verify this for k ≤ C log n with any fixed C > 0. In a companion paper, using very different methods, we treat the complementary range, proving the conjecture for k ≥ κ 0 log n (with κ 0 ≈ 4.82).

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C

such that for every

n

and

k | n

, the random graph

G (n, C \frac{\log n}{n})

w.h.p.…”

Section: Introductionmentioning

confidence: 92%

“…While this does not so far seem to be leading to a proof of Conjecture , it is plausible that the methods of paper at least extend to any (bounded‐degree) tree with

o (\sqrt{n})

leaves.…”

Section: Introductionmentioning

confidence: 99%

The threshold for combs in random graphs

Lubetzky

Wormald

2015

Random Struct. Alg.

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“…The first spanning structures considered in graphs were perfect matchings and Hamilton cycles (see also , Chapter 8] and references therein). More recently, the thresholds for the appearance of (bounded degree) spanning trees were studied as well, for the currently best bounds see .…”

Section: Introductionmentioning

confidence: 99%

Spanning structures and universality in sparse hypergraphs

Parczyk

Person

2016

Random Struct Algorithms

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Abstract. In this paper the problem of finding various spanning structures in random hypergraphs is studied. We notice that a general result of Riordan [Spanning subgraphs of random graphs, Combinatorics, Probability & Computing 9 (2000), no. 2, 125-148] can be adapted from random graphs to random r-uniform hypergaphs and provide sufficient conditions when a random r-uniform hypergraph H (r) (n, p) contains a given spanning structure a.a.s. We also discuss several spanning structures such as cube-hypergraphs, lattices, spheres and Hamilton cycles in hypergraphs.Moreover, we study universality, i.e. when does an r-uniform hypergraph contain any hypergraph on n vertices and with maximum vertex degree bounded by ∆? For H (r) (n, p) it is shown that this holds for p = ω (ln n/n) 1/∆ a.a.s. by combining approaches taken by Dellamonica, Kohayakawa, Rödl and Ruciński [An improved upper bound on the density of universal random graphs, Random Structures Algorithms 46 (2015), no. 2, 274-299] and of Ferber, Nenadov and Peter [Universality of random graphs and rainbow embedding, Random Structures Algorithms, to appear]. Furthermore it is shown that the random graph G(n, p) for appropriate p and explicit constructions due to Alon, Capalbo, Kohayakawa, Rödl, Ruciński and Szemerédi [7] and Alon and Capalbo [4,5] of universal graphs yield constructions of universal hypergraphs that are sparser than the random hypergraph H (r) (n, p) with p = ω (ln n/n) 1/∆ .

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“…Observe that, for example for k = √ n, combs neither have linearly many leaves nor linear sized bare paths. Kahn, Lubetzky, and Wormald [84,83] established Conjecture 4.6 for combs. This was improved on and generalised by Montgomery [114] who proved the following result.…”

Section: Treesmentioning

confidence: 99%

Large-scale structures in random graphs

Böttcher

2017

Surveys in Combinatorics 2017

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In recent years there has been much progress in graph theory on questions of the following type. What is the threshold for a certain large substructure to appear in a random graph? When does a random graph contain all structures from a given family? And when does it contain them so robustly that even an adversary who is allowed to perturb the graph cannot destroy all of them? I will survey this progress, and highlight the vital role played by some newly developed methods, such as the sparse regularity method, the absorbing method, and the container method. I will also mention many open questions that remain in this area.

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Cycle Factors and Renewal Theory

Cited by 6 publications

References 29 publications

The threshold for combs in random graphs

The threshold for combs in random graphs

Spanning structures and universality in sparse hypergraphs

Large-scale structures in random graphs

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