Hamiltonicity of graphs perturbed by a random geometric graph

Díaz, Alberto Espuny

doi:10.1002/jgt.22901

Cited by 5 publications

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“…We believe it would be interesting to study graphs perturbed by a random graph with a fixed degree sequence as well. Very recently, the first author has also considered Hamiltonicity of graphs perturbed by a random geometric graph [18]. Of course, this study should also be extended to other graph properties.…”

Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

Hamiltonicity of graphs perturbed by a random regular graph

Díaz

Girão

2022

Random Struct Algorithms

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We study Hamiltonicity and pancyclicity in the graph obtained as the union of a deterministic n$$ n $$‐vertex graph H$$ H $$ with δfalse(Hfalse)≥αn$$ \delta (H)\ge \alpha n $$ and a random d$$ d $$‐regular graph G$$ G $$, for d∈false{1,2false}$$ d\in \left\{1,2\right\} $$. When G$$ G $$ is a random 2‐regular graph, we prove that a.a.s. H∪G$$ H\cup G $$ is pancyclic for all α∈false(0,1false]$$ \alpha \in \left(0,1\right] $$, and also extend our result to a range of sublinear degrees. When G$$ G $$ is a random 1‐regular graph, we prove that a.a.s. H∪G$$ H\cup G $$ is pancyclic for all α∈false(2prefix−1,1false]$$ \alpha \in \left(\sqrt{2}-1,1\right] $$, and this result is best possible. Furthermore, we show that this bound on δfalse(Hfalse)$$ \delta (H) $$ is only needed when H$$ H $$ is “far” from containing a perfect matching, as otherwise we can show results analogous to those for random 2‐regular graphs. Our proofs provide polynomial‐time algorithms to find cycles of any length.

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Section: Concluding Remarks and Open Problemsmentioning

confidence: 99%

Hamiltonicity of graphs perturbed by a random regular graph

Díaz

Girão

2022

Random Struct Algorithms

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“…The study of the emergence of various spanning configurations in randomly perturbed (hyper)graphs dates back to the work of Bohman et al [12], and received significant attention in recent years, see for example, [5,9–11,14,15,17,27,28,30–32] to name just a few. In most results in this area, the random perturbation is the binomial random graph

𝔾 false(n, p false)

(or the, essentially equivalent, Erdős‐Rényi random graph

𝔾 false(n, m false)

); however, quite recently other distributions have been considered, such as random geometric graphs [19] and random regular graphs [16]. In the context of rainbow spanning configurations, the emergence of rainbow Hamilton cycles in uniformly colored randomly perturbed graphs was studied by Anastos and Frieze [4] as well as by the first two authors [1].…”

Section: Introductionmentioning

confidence: 99%

Rainbow trees in uniformly edge‐colored graphs

Aigner-Horev

Hefetz

Lahiri

2022

Random Struct Algorithms

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We obtain sufficient conditions for the emergence of spanning and almost-spanning bounded-degree rainbow trees in various host graphs, having their edges colored independently and uniformly at random, using a predetermined palette. Our first result asserts that a uniform coloring of G(n, 𝜔(1)∕n), using a palette of size n, a.a.s. admits a rainbow copy of any given bounded-degree tree on at most (1 − 𝜀)n vertices, where 𝜀 > 0 is arbitrarily small yet fixed. This serves as a rainbow variant of a classical result by Alon et al. pertaining to the embedding of bounded-degree almost-spanning prescribed trees in G(n, C∕n), where C > 0 is independent of n. Given an n-vertex graph G with minimum degree at least 𝛿n, where 𝛿 > 0 is fixed, we use our aforementioned result in order to prove that a uniform coloring of the randomly perturbed graph G ∪ G(n, 𝜔(1)∕n), using (1 + 𝛼)n colors, where 𝛼 > 0 is arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given bounded-degree spanning tree. This can be viewed as a rainbow variant of a result by Krivelevich et al. who proved that G∪G(n, C∕n), where C > 0 is independent of n, a.a.s. admits a copy of any given bounded-degree spanning tree. Finally, and with G as above, we prove that a uniform coloring of G∪G(n, 𝜔(n −2 )) using n−1 colors a.a.s. admits a rainbow spanning tree. Put another way, the trivial lower bound on the size of the palette required for supporting a rainbow spanning tree is also sufficient, essentially as soon as the random perturbation a.a.s. has edges.

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“…At this point, one may consider an auxiliary path which visits every cell once and construct a Hamilton cycle by traversing the cells twice following the path (forwards and then backwards), incorporating the vertices into the cycle while doing so. Technical variations of this idea have been used to obtain many results about Hamilton cycles in random geometric graphs [4,5,18,19,20,23,26,37,40,48].…”

Section: B ( ) Hmentioning

confidence: 99%

“…Therefore, when considering this problem, we must really consider all possible values of . Second, all known results about Hamiltonicity in ( , ) exploit the fact that these graphs are locally dense, meaning that vertices which are (geometrically) close form very large cliques [4,5,18,19,20,23,26,37,40,48]. This is not the case in the local resilience setting, where the adversary may delete all cliques.…”

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A. Given an increasing graph property P, a graph is -resilient with respect to P if, for every spanning subgraph ⊆ where each vertex keeps more than a (1 − )-proportion of its neighbours, has property P. We study the above notion of local resilience with being a random geometric graph ( , ) obtained by embedding vertices independently and uniformly at random in [0, 1] , and connecting two vertices by an edge if the distance between them is at most .First, we focus on connectivity. We show that, for every > 0, for a constant factor above the sharp threshold for connectivity of ( , ), the random geometric graph is (1/2 − )-resilient for the property of being -connected, with of the same order as the expected degree. However, contrary to binomial random graphs, for sufficiently small > 0, connectivity is not born (1/2 − )-resilient in 2-dimensional random geometric graphs.Second, we study local resilience with respect to the property of containing long cycles. We show that, for a constant factor above , ( , ) is (1/2 − )-resilient with respect to containing cycles of all lengths between constant and 2 /3. Proving (1/2 − )-resilience for Hamiltonicity remains elusive with our techniques. Nevertheless, we show that ( , ) is -resilient with respect to Hamiltonicity for a fixed constant = ( ) < 1/2. This result can be regarded as a version of Dirac's theorem for binomial random graphs. Recently, Montgomery [38] and Nenadov, Steger and Trujić [42] independently sharpened this result to a "hitting time" result. Moreover, Theorem 1.1 was recently generalised by Condon, Espuny Díaz, Kim, Kühn and Osthus [14], who considered a version of the problem where the adversary is allowed to delete more edges incident to some, but not all, vertices of the graph, giving an analogue of a classical result of Pósa [50] for random graphs.The local resilience with respect to Hamiltonicity has also been studied in other models of random graphs. One example is the model of random regular graphs. Given integers 1 ≤ < , a random -regular graph , is obtained by sampling an element from the set of all -vertex -regular graphs uniformly at random. It is well known that the random -regular graph is a.a.s. Hamiltonian for all ≥ 3 [16, 35, 51, 52]. Improving on earlier work of Ben-Shimon, Krivelevich and Sudakov [8], Condon, Espuny Díaz, Girão, Kühn and Osthus [13] proved a result analogous to Theorem 1.1 for random regular graphs.Theorem 1.2 ([13, Theorem 1.2]). For every ∈ (0, 1/2], there exists a constant = ( ) > 0 such that, for all ∈ [ , − 1], a.a.s. every (1/2 + )-subgraph of , is Hamiltonian.Just like for binomial random graphs, the constant 1/2 is best possible. Moreover, Condon, Espuny Díaz, Girão, Kühn and Osthus [13] also showed that a polynomial dependency between and in Theorem 1.2 cannot be avoided.1 We remark that our definition of local resilience is slightly different from that in [55] and closer to more recent versions which have appeared in the literature [38, 42]. Nevertheless, all the results we mention in our paper hold with ...

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Hamiltonicity of graphs perturbed by a random geometric graph

Cited by 5 publications

References 28 publications

Hamiltonicity of graphs perturbed by a random regular graph

Hamiltonicity of graphs perturbed by a random regular graph

Rainbow trees in uniformly edge‐colored graphs

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