Robustness of graph properties

Sudakov, Benny

doi:10.1017/9781108332699.009

Cited by 30 publications

(30 citation statements)

References 91 publications

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“…Even though the study of resilience properties of random graphs has attracted considerable attention over the last years, cf. for example and the recent survey , this is the first result which determines precisely when the random graph process becomes resilient with respect to some well‐studied property.…”

Section: Introductionmentioning

confidence: 80%

Resilience of perfect matchings and Hamiltonicity in random graph processes

Nenadov

Steger

Trujić

2018

Random Struct Algorithms

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Let {Gi} be the random graph process: starting with an empty graph G0 with n vertices, in every step i ≥ 1 the graph Gi is formed by taking an edge chosen uniformly at random among the nonexisting ones and adding it to the graph Gi − 1. The classical “hitting‐time” result of Ajtai, Komlós, and Szemerédi, and independently Bollobás, states that asymptotically almost surely the graph becomes Hamiltonian as soon as the minimum degree reaches 2, that is if δ(Gi) ≥ 2 then Gi is Hamiltonian. We establish a resilience version of this result. In particular, we show that the random graph process almost surely creates a sequence of graphs such that for m≥false(16+ofalse(1false)false)nnormallogn edges, the 2‐core of the graph Gm remains Hamiltonian even after an adversary removes false(12−ofalse(1false)false)‐fraction of the edges incident to every vertex. A similar result is obtained for perfect matchings.

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Section: Introductionmentioning

confidence: 80%

Resilience of perfect matchings and Hamiltonicity in random graph processes

Nenadov

Steger

Trujić

2018

Random Struct Algorithms

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“…a Dirac-type generalisation of [5]). The local resilience perspective emphasises analogies with the recent literature on Dirac-type problems in the random setting (see the surveys [1,26]), perhaps suggests looking for common generalisations, e.g. a rainbow version of [18]: in the random graph G(n, p) with p > C(log n)/n, must any o(pn)-bounded edge-colouring of any subgraph H with minimum degree (1/2 + o(1))pn have a rainbow Hamilton cycle?…”

Section: Discussionmentioning

confidence: 92%

Rainbow factors in hypergraphs

Coulson

Keevash

Perarnau

et al. 2020

Journal of Combinatorial Theory, Series A

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For any r-graph H, we consider the problem of finding a rainbow H-factor in an r-graph G with large minimum ℓ-degree and an edge-colouring that is suitably bounded. We show that the asymptotic degree threshold is the same as that for finding an H-factor. IntroductionA fundamental question in Extremal Combinatorics is to determine conditions on a hypergraph G that guarantee an embedded copy of some other hypergraph H. The Turán problem for an r-graph H asks for the maximum number of edges in an r-graph G on n vertices; we usually think of H as fixed and n as large. For r = 2 (ordinary graphs) this problem is fairly well understood (except when H is bipartite), but for general r and general H we do not even have an asymptotic understanding of the Turán problem (see the survey [11]). For example, a classical theorem of Mantel determines the maximum number of edges in a triangle-free graph on n vertices (it is n 2 /4 ), but we do not know even asymptotically the maximum number of edges in a tetrahedron-free 3-graph on n vertices. On the other hand, if we seek to embed a spanning hypergraph then it is most natural to consider minimum degree conditions. Such questions are known as Dirac-type problems, after the classical theorem of Dirac that any graph on n ≥ 3 vertices with minimum degree at least n/2 contains a Hamilton cycle. There is a large literature on such problems for graphs and hypergraphs, surveyed in [15,17,22,28].One of these problems, finding hypergraph factors, will be our topic for the remainder of this paper. To describe it we introduce some notation and terminology. Let G be an r-graph on [n] = {1, . . . , n}. For any L ⊆ V (G) the degree of L in G is the number of edges of G containing L. The minimum ℓ-degree δ ℓ (G) is the minimum degree in G over all L ⊆ V (G) of size ℓ. Let H be an r-graph with |V (H)| = h | n. A partial H-factor F in G of size m is a set of m vertex-disjoint copies of H in G. If m = n/h we call F an H-factor. We let δ ℓ (H, n) be the minimum δ such that δ ℓ (G) > δn r−ℓ guarantees an H-factor in G. Then the asymptotic ℓ-degree threshold for H-factors is δ * ℓ (H) := lim inf m→∞ δ ℓ (H, mh) .

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“…Forbidden transitions can also be used to measure the robustness of graph properties. In [10], Sudakov studies the Hamiltonicity of a graph with the idea that even Hamiltonian graphs can be more or less strongly Hamiltonian (an Hamiltonian graph is a graph in which there exists an elementary cycle that uses all the vertices). The number of transitions one needs to forbid for a graph to lose its Hamiltonicity gives a measure of its robustness: if the smallest set of forbidden transitions that makes a graph lose its Hamiltonicity has size 4, this means that this graph can withstand the failure of three transitions, no matter where the failures happen.…”

Section: Introductionmentioning

confidence: 99%

On Minimum Connecting Transition Sets in Graphs

Bellitto

Bergougnoux

2018

Graph-Theoretic Concepts in Computer Science

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A forbidden transition graph is a graph defined together with a set of permitted transitions i.e. unordered pair of adjacent edges that one may use consecutively in a walk in the graph. In this paper, we look for the smallest set of transitions needed to be able to go from any vertex of the given graph to any other. We prove that this problem is NP-hard and study approximation algorithms. We develop theoretical tools that help to study this problem.

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Robustness of graph properties

Cited by 30 publications

References 91 publications

Resilience of perfect matchings and Hamiltonicity in random graph processes

Resilience of perfect matchings and Hamiltonicity in random graph processes

Rainbow factors in hypergraphs

On Minimum Connecting Transition Sets in Graphs

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