On powers of tight Hamilton cycles in randomly perturbed hypergraphs

Chang, Yuhua; Han, Jie; Thoma, Luboš

doi:10.1002/rsa.21154

Cited by 2 publications

(1 citation statement)

References 55 publications

(155 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The model of randomly perturbed graphs has also been extended to other settings. For instance, Hamiltonicity has been studied in randomly perturbed directed graphs [5, 24], hypergraphs [4, 12, 21, 24, 27] and subgraphs of the hypercube [13]. A common phenomenon in randomly perturbed graphs is that, by considering the union with a dense graph (i.e., with linear degrees), the threshold for the probabilities of different properties is significantly lower than that of the classical

{G}_{n,p}

model.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonicity of graphs perturbed by a random regular graph

Díaz

Girão

2022

Random Struct Algorithms

View full text Add to dashboard Cite

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.

show abstract

{G}_{n,p}

model.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonicity of graphs perturbed by a random regular graph

Díaz

Girão

2022

Random Struct Algorithms

View full text Add to dashboard Cite

show abstract

Compatible powers of Hamilton cycles in dense graphs

Cheng,

Hu,

Yang

2024

Journal of Graph Theory

View full text Add to dashboard Cite

The notion of incompatibility system was first proposed by Krivelevich, Lee and Sudakov to formulate the robustness of Hamiltonicity of Dirac graphs. Given a graph , an incompatibility system over is a family such that for every , is a family of edge pairs with . Moreover, for an integer , we say is ‐bounded if for every vertex and its incident edge , there are at most pairs in containing . Krivelevich, Lee and Sudakov proved that there is an universal constant such that for every Dirac graph and every ‐bounded incompatibility system over , there exists a Hamilton cycle where every pair of adjacent edges of satisfies for . This resolves a conjecture posed by Häggkvist in 1988 and such a Hamilton cycle is called compatible (with respect to ). We study high powers of Hamilton cycles in this context and show that for every and , there exists a constant such that for sufficiently large and every ‐bounded incompatibility system over an ‐vertex graph with , there exists a compatible th power of a Hamilton cycle in . Moreover, we give a ‐bounded construction which has minimum degree and contains no compatible th power of a Hamilton cycle.

show abstract

On powers of tight Hamilton cycles in randomly perturbed hypergraphs

Cited by 2 publications

References 55 publications

Hamiltonicity of graphs perturbed by a random regular graph

Hamiltonicity of graphs perturbed by a random regular graph

Compatible powers of Hamilton cycles in dense graphs

Contact Info

Product

Resources

About