Hamilton cycles in dense vertex-transitive graphs

Christofides, Demetres; Hladký, Jan; Máthé, András

doi:10.1016/j.jctb.2014.05.001

Cited by 12 publications

(11 citation statements)

References 26 publications

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“…Given a graph G and a partition of V (G) into clusters of equal size, let R d (G) be the graph whose vertices correspond to clusters and where an edge indicates that the bipartite graph induced by the corresponding clusters is ε-regular with density at least d. If there is a connected component of R d (G) which contains a matching covering at least (α + d)v(R d (G)) of the vertices of R d (G), then there are paths and even cycles in G of each length up to αv(G). In fact, a stronger result is true, as observed by Hladký, Krá ' l and Piguet (see [6], where this idea was also used): we need only a fractional matching with weight (α + d)v(R d (G)). In this subsection we state the corresponding result for tight paths and cycles in k-graphs.…”

Section: Cycle Embedding Lemmamentioning

confidence: 94%

Tight cycles and regular slices in dense hypergraphs

Allen

Böttcher

Cooley

et al. 2017

Journal of Combinatorial Theory, Series A

277

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Abstract. We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of the original hypergraph. Accordingly we advocate their use in extremal hypergraph theory, and explain how they can lead to considerable simplifications in existing proofs in this field. We also use them for establishing the following two new results.Firstly, we prove a hypergraph extension of the Erdős-Gallai Theorem: for every δ > 0 every sufficiently large k-uniform hypergraph with at least (α + δ) n k edges contains a tight cycle of length αn for each α ∈ [0, 1].Secondly, we find (asymptotically) the minimum codegree requirement for a k-uniform k-partite hypergraph, each of whose parts has n vertices, to contain a tight cycle of length αkn, for each 0 < α < 1.

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Section: Cycle Embedding Lemmamentioning

confidence: 94%

Tight cycles and regular slices in dense hypergraphs

Allen

Böttcher

Cooley

et al. 2017

Journal of Combinatorial Theory, Series A

277

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“…In the special case of dense vertex-transitive graphs (which are always regular), Christofides, Hladký and Máthé [7] introduced a partition into 'iron connected components'. (Iron connectivity is closely related to robust expansion.)…”

Section: Introductionmentioning

confidence: 99%

The robust component structure of dense regular graphs and applications

Kühn

Osthus

et al. 2014

Proceedings of the London Mathematical Society

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In this paper, we study the large‐scale structure of dense regular graphs. This involves the notion of robust expansion, a recent concept which has already been used successfully to settle several longstanding problems. Roughly speaking, a graph is robustly expanding if it still expands after the deletion of a small fraction of its vertices and edges. Our main result allows us to harness the useful consequences of robust expansion even if the graph itself is not a robust expander. It states that every dense regular graph can be partitioned into ‘robust components’, each of which is a robust expander or a bipartite robust expander. We apply our result to obtain (amongst others) the following. We prove that whenever ε>0, every sufficiently large 3‐connected D‐regular graph on n vertices with D⩾(14+ε)n is Hamiltonian. This asymptotically confirms the only remaining case of a conjecture raised independently by Bollobás and Häggkvist in the 1970s. We prove an asymptotically best possible result on the circumference of dense regular graphs of given connectivity. The 2‐connected case of this was conjectured by Bondy and proved by Wei.

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“…As already mentioned, the method introduced by Luczak allows us to lift large connected matchings in a reduced graph R i to large cycles in the corresponding G i . In fact, Christofides et al [12] observed that the same is also true for fractional matchings and similarly, 2-matchings. Formally, the following statement holds:…”

Section: Regularitymentioning

confidence: 75%

Ore- and Pósa-type conditions for partitioning $2$-edge-coloured graphs into monochromatic cycles

Patrick¹

2022

Preprint

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In 2019, Letzter confirmed a conjecture of Balogh, Barát, Gerbner, Gyárfás and Sárközy, proving that every large 2-edge-coloured graph G on n vertices with minimum degree at least 3n/4 can be partitioned into two monochromatic cycles of different colours. Here, we propose a weaker condition on the degree sequence of G to also guarantee such a partition and prove an approximate version. Continuing work by Allen, Böttcher, Lang, Skokan and Stein, we also show that if deg(u) + deg(v) 4n/3 + o(n) holds for all non-adjacent vertices u, v ∈ V (G), then all but o(n) vertices can be partitioned into three monochromatic cycles.

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Hamilton cycles in dense vertex-transitive graphs

Cited by 12 publications

References 26 publications

Tight cycles and regular slices in dense hypergraphs

Tight cycles and regular slices in dense hypergraphs

The robust component structure of dense regular graphs and applications

Ore- and Pósa-type conditions for partitioning $2$-edge-coloured graphs into monochromatic cycles

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