Long Cycles in Locally Expanding Graphs, with Applications

Krivelevich, Michael

doi:10.1007/s00493-017-3701-1

Cited by 37 publications

(46 citation statements)

References 39 publications

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“…The following lemma (whose proof is almost identical to that of Proposition 3 of ) verifies that random regular graphs satisfy the “local sparseness” condition (ii) of Theorem…”

Section: Proof Of Theorems 11(ii) and 12(ii)mentioning

confidence: 71%

“…On the other hand, they prove that there exist r 2 -regular graphs on n vertices such that every edge r-coloring contains a monochromatic cycle of length at least n Ω(log ). Our first theorem provides an analog of the results of [3,11,14] in the setting of random d-regular graphs, n d ( , ).…”

Section: Introductionmentioning

confidence: 81%

“…This section follows closely the recent paper of Krivelevich . We begin by stating a theorem from which we will use.…”

Section: Proof Of Theorems 11(ii) and 12(ii)mentioning

confidence: 94%

“…This section follows closely the recent paper of Krivelevich . We begin by stating a theorem from which we will use. This theorem says that graphs with decent global density, but relatively smaller local density must contain a long cycle.…”

Section: Proof Of Theorems 11(ii) and 12(ii)mentioning

confidence: 94%

“…The constant ψ r actually arises from the so-called r-orientability threshold which was discovered independently by Cain, Sanders, and Wormald [5] and by Fernholz and Ramachandran [6]. A graph is called r-orientable if its edges can be oriented such that the maximum in-degree of any vertex is at most r. Recently, Krivelevich [11] improved upon the results in [3,14] by showing that w.h.p. every edge r-coloring of G n ψ ε n ( , ( + )/ ) r contains not only a linear-sized monochromatic component, but also actually a linear length monochromatic cycle.…”

Section: Introductionmentioning

confidence: 99%

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A Ramsey property of random regular and k‐out graphs

Anastos

Bal

2019

Journal of Graph Theory

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In this study we consider a Ramsey property of random d‐regular graphs, G(n,d). Let rgoodbreakinfix≥2 be fixed. Then w.h.p. the edges of G(n,2r) can be colored such that every monochromatic component has order o(n). On the other hand, there exists a constant γgoodbreakinfix>0 such that w.h.p., every r‐coloring of the edges of G(n,2r+1) must contain a monochromatic cycle of length at least γn. We prove an analogous result for random k‐out graphs.

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“…The following lemma (whose proof is almost identical to that of Proposition 3 of ) verifies that random regular graphs satisfy the “local sparseness” condition (ii) of Theorem…”

Section: Proof Of Theorems 11(ii) and 12(ii)mentioning

confidence: 71%

Section: Introductionmentioning

confidence: 81%

“…This section follows closely the recent paper of Krivelevich . We begin by stating a theorem from which we will use.…”

Section: Proof Of Theorems 11(ii) and 12(ii)mentioning

confidence: 94%

Section: Proof Of Theorems 11(ii) and 12(ii)mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Ramsey property of random regular and k‐out graphs

Anastos

Bal

2019

Journal of Graph Theory

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The size‐Ramsey number of short subdivisions

Draganić

Krivelevich

Nenadov

2021

Random Struct Algorithms

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The r-size-Ramsey numberR r (H) of a graph H is the smallest number of edges a graph G can have such that for every edge-coloring of G with r colors there exists a monochromatic copy of H in G. For a graph H, we denote by H q the graph obtained from H by subdividing its edges with q − 1 vertices each. In a recent paper of Kohayakawa, Retter and Rödl, it is shown that for all constant integers q, r ≥ 2 and every graph H on n vertices and of bounded maximum degree, the r-size-Ramsey number of H q is at most (log n) 20(q−1) n 1+1∕q , for n large enough. We improve upon this result using a significantly shorter argument by showing thatR r (H q ) ≤ O(n 1+1∕q ) for any such graph H.

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Isoperimetric Inequalities and Supercritical Percolation on High-Dimensional Graphs

Diskin,

Erde,

Kang

et al. 2024

Combinatorica

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It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component $$L_1$$ L 1 in supercritical percolation on these product graphs, that is, when $$p=\frac{1+\epsilon }{d}$$ p = 1 + ϵ d , where d is the degree of the product graph and $$\epsilon >0$$ ϵ > 0 is a small enough constant. We show that typically $$L_1$$ L 1 has edge-expansion $$\Omega \left( \frac{1}{d\ln d}\right) $$ Ω 1 d ln d . Furthermore, we show that $$L_1$$ L 1 likely contains a linear-sized subgraph with vertex-expansion $$\Omega \left( \frac{1}{d\ln d}\right) $$ Ω 1 d ln d . These results are best possible up to the logarithmic factor in d. Using these likely expansion properties, we determine, up to small polylogarithmic factors in d, the likely diameter of $$L_1$$ L 1 as well as the typical mixing time of a lazy random walk on $$L_1$$ L 1 . Furthermore, we show the likely existence of a cycle of length $$\Omega \left( \frac{n}{d\ln d}\right) $$ Ω n d ln d . These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of $$L_1$$ L 1 were previously only known to be polynomial in d.

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Long Cycles in Locally Expanding Graphs, with Applications

Abstract: We provide sufficient conditions for the existence of long cycles in locally expanding graphs, and present applications of our conditions and techniques to Ramsey theory, random graphs and positional games.

Cited by 37 publications

References 39 publications

A Ramsey property of random regular and k‐out graphs

A Ramsey property of random regular and k‐out graphs

The size‐Ramsey number of short subdivisions

Isoperimetric Inequalities and Supercritical Percolation on High-Dimensional Graphs

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