A Degree Sequence Strengthening of the Vertex Degree Threshold for a Perfect Matching in 3-Uniform Hypergraphs

Bowtell, Candida; Hyde, Joseph

doi:10.1137/20m1364825

Cited by 3 publications

(4 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For uniform linear hypergraphs, Jiang and Ma [19] settled a conjecture of Verstraëte, by finding asymptotic minimum degree condition necessary for the existence of Berge cycles of k consecutive lengths. For the study of asymptotic minimum degree thresholds that force matching see the following work [4,16]. Katona and Kierstead [20] introduced an alternative definition of Hamiltonian cycles in hypergraphs, a notion that has garnered considerable attention over recent decades.…”

Section: Hamiltonicity For Hypergraphs and Main Resultsmentioning

confidence: 99%

Pósa-Type Results for Berge Hypergraphs

Salia

2024

Electron. J. Combin.

View full text Add to dashboard Cite

A Berge cycle of length $k$ in a hypergraph $\mathcal H$ is a sequence of distinct vertices and hyperedges $v_1,h_1,v_2,h_2,\dots,v_{k},h_k$ such that $v_{i},v_{i+1}\in h_i$ for all $i\in[k]$, indices taken modulo $k$. F\"uredi, Kostochka and Luo recently gave sharp Dirac-type minimum degree conditions that force non-uniform hypergraphs to have Hamiltonian Berge cycles. We give a sharp Pósa-type lower bound for $r$-uniform and non-uniform hypergraphs that force Hamiltonian Berge cycles.

show abstract

Section: Hamiltonicity For Hypergraphs and Main Resultsmentioning

confidence: 99%

Pósa-Type Results for Berge Hypergraphs

Salia

2024

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…For example, Treglown [22] gave a degree sequence condition that forces the graph to contain a clique factor and Staden and Treglown [20] proved a degree sequence condition that forces the graph to contain the square of a Hamiltonian cycle. Since the first version of this article, Bowtell and Hyde [1] obtained a degree sequence condition for perfect matchings in 3-graphs.…”

Section: Theorem 14 (Main Result)mentioning

confidence: 99%

“…In recent years, there has been some progress to achieve Dirac like results for hypergraphs. Rödl, Ruciński, and Szemerédi [18] started by showing that for α > 0, there is some n 0 such that every 3-graph on n ≥ n 0 vertices with minimum pair degree at least 1 2 + α n contains a Hamiltonian cycle. Actually, in [19] they improved the result to the following.…”

Section: Introductionmentioning

confidence: 99%

A pair degree condition for Hamiltonian cycles in 3-uniform hypergraphs

Schülke

2023

Combinator. Probab. Comp.

View full text Add to dashboard Cite

We prove a new sufficient pair degree condition for tight Hamiltonian cycles in $3$ -uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to Rödl, Ruciński, and Szemerédi. For graphs, Chvátal characterised all those sequences of integers for which every pointwise larger (or equal) degree sequence guarantees the existence of a Hamiltonian cycle. A step towards Chvátal’s theorem was taken by Pósa, who improved on Dirac’s tight minimum degree condition for Hamiltonian cycles by showing that a certain weaker condition on the degree sequence of a graph already yields a Hamiltonian cycle. In this work, we take a similar step towards a full characterisation of all pair degree matrices that ensure the existence of tight Hamiltonian cycles in $3$ -uniform hypergraphs by proving a $3$ -uniform analogue of Pósa’s result. In particular, our result strengthens the asymptotic version of the result by Rödl, Ruciński, and Szemerédi.

show abstract

“…Finally, it is natural to ask whether these types of results generalise to hypergraphs. On the practical side, first steps in this direction were undertaken by Schülke [69] for tight cycles under codegree sequence conditions in 3‐uniform hypergraphs and by Bowtell and Hyde [9] for perfect matchings under minimum vertex degree sequence conditions in 3‐uniform hypergraphs. On the theoretical side, the existence of Hamilton frameworks minimum degree conditions for tight Hamilton cycles has been proposed in our earlier work [54].…”

Section: Discussionmentioning

confidence: 99%

On sufficient conditions for spanning structures in dense graphs

Lang

Sanhueza‐Matamala

2023

Proceedings of London Math Soc

View full text Add to dashboard Cite

We study structural conditions in dense graphs that guarantee the existence of vertex‐spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. A simple consequence of the Robust Expander Theorem of Kühn, Osthus and Treglown tells us that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Our main result generalises this phenomenon to powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles. This answers a question of Ebsen, Maesaka, Reiher, Schacht and Schülke and solves the embedding problem that underlies multiple lines of research on sufficient conditions for spanning structures in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore‐type degree conditions, Pósa‐type degree conditions, deficiency‐type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders.

show abstract

A Degree Sequence Strengthening of the Vertex Degree Threshold for a Perfect Matching in 3-Uniform Hypergraphs

Cited by 3 publications

References 21 publications

Pósa-Type Results for Berge Hypergraphs

Pósa-Type Results for Berge Hypergraphs

A pair degree condition for Hamiltonian cycles in 3-uniform hypergraphs

On sufficient conditions for spanning structures in dense graphs

Contact Info

Product

Resources

About