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Embedding loose spanning trees in 3-uniform hypergraphs
Published 23 Jan 2023 in math.CO | (2301.09630v3)
Abstract: In 1995, Koml\'os, S\'ark\"ozy and Szemer\'edi showed that every large $n$-vertex graph with minimum degree at least $(1/2 + \gamma)n$ contains all spanning trees of bounded degree. We consider a generalization of this result to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained by successively appending edges sharing a single vertex with a previous edge. We show that for all $\gamma$ and $\Delta$, and $n$ large, every $n$-vertex 3-uniform hypergraph of minimum vertex degree $(5/9 + \gamma)\binom{n}{2}$ contains every loose spanning tree $T$ with maximum vertex degree $\Delta$. This bound is asymptotically tight, since some loose trees contain perfect matchings.
- Universality for bounded degree spanning trees in randomly perturbed graphs. Random Structures Algorithms, 55(4):854–864, 2019.
- Embedding spanning bounded degree graphs in randomly perturbed graphs. Mathematika, 66(2):422–447, 2020.
- Minimum vertex degree conditions for loose Hamilton cycles in 3-uniform hypergraphs. J. Combin. Theory Ser. B, 103(6):658–678, 2013.
- F. R. K. Chung. Regularity lemmas for hypergraphs and quasi-randomness. Random Structures Algorithms, 2(2):241–252, 1991.
- Tight bounds for embedding bounded degree trees. In Fete of combinatorics and computer science, volume 20 of Bolyai Soc. Math. Stud., pages 95–137. János Bolyai Math. Soc., Budapest, 2010.
- P. Frankl and V. Rödl. The uniformity lemma for hypergraphs. Graphs Combin., 8(4):309–312, 1992.
- On perfect matchings in uniform hypergraphs with large minimum vertex degree. SIAM J. Discrete Math., 23(2):732–748, 2009.
- J. Han and Y. Zhao. Minimum vertex degree threshold for loose Hamilton cycles in 3-uniform hypergraphs. J. Combin. Theory Ser. B, 114:70–96, 2015.
- G. Kalai. Enumeration of 𝐐𝐐{\bf Q}bold_Q-acyclic simplicial complexes. Israel J. Math., 45(4):337–351, 1983.
- A. Kathapurkar and R. Montgomery. Spanning trees in dense directed graphs. J. Combin. Theory Ser. B, 156:223–249, 2022.
- I. Khan. Perfect matchings in 3-uniform hypergraphs with large vertex degree. SIAM J. Discrete Math., 27(2):1021–1039, 2013.
- Proof of a packing conjecture of Bollobás. Combin. Probab. Comput., 4(3):241–255, 1995.
- Spanning trees in dense graphs. Combin. Probab. Comput., 10(5):397–416, 2001.
- Matchings in 3-uniform hypergraphs. J. Combin. Theory Ser. B, 103(2):291–305, 2013.
- R. Lang and N. Sanhueza-Matamala. Minimum degree conditions for tight Hamilton cycles. J. Lond. Math. Soc. (2), 105(4):2249–2323, 2022.
- N. Linial and Y. Peled. Enumeration and randomized constructions of hypertrees. Random Structures Algorithms, 55(3):677–695, 2019.
- R. Mycroft and T. Naia. Spanning trees of dense directed graphs. In The proceedings of Lagos 2019, the tenth Latin and American Algorithms, Graphs and Optimization Symposium (LAGOS 2019), volume 346 of Electron. Notes Theor. Comput. Sci., pages 645–654. Elsevier Sci. B. V., Amsterdam, 2019.
- R. Mycroft and T. Naia. Trees and tree-like structures in dense digraphs. arXiv e-prints, Dec. 2020, arXiv:2012.09201.
- Dirac-type conditions for spanning bounded-degree hypertrees. Journal of Combinatorial Theory, Series B, 165:97–141, 2024.
- Minimum vertex degree condition for tight Hamiltonian cycles in 3-uniform hypergraphs. Proc. Lond. Math. Soc. (3), 119(2):409–439, 2019.
- K. F. Roth. On certain sets of integers. J. London Math. Soc., 28:104–109, 1953.
- M. Stein. Tree containment and degree conditions. In Discrete mathematics and applications, volume 165 of Springer Optim. Appl., pages 459–486. Springer, Cham, [2020] ©2020.
- E. Szemerédi. On sets of integers containing no k𝑘kitalic_k elements in arithmetic progression. Acta Arith., 27:199–245, 1975.
- E. Szemerédi. Regular partitions of graphs. In Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), volume 260 of Colloq. Internat. CNRS, pages 399–401. CNRS, Paris, 1978.
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