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Recent advances on Dirac-type problems for hypergraphs

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Recent Trends in Combinatorics

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 159))

Abstract

A fundamental question in graph theory is to establish conditions that ensure a graph contains certain spanning subgraphs. Two well-known examples are Tutte’s theorem on perfect matchings and Dirac’s theorem on Hamilton cycles. Generalizations of Dirac’s theorem, and related matching and packing problems for hypergraphs, have received much attention in recent years. New tools such as the absorbing method and regularity method have helped produce many new results, and yet some fundamental problems in the area remain unsolved. We survey recent developments on Dirac-type problems along with the methods involved, and highlight some open problems.

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Notes

  1. 1.

    We are not aware of any packing results on d-degree conditions for 1 < d < k − 1.

  2. 2.

    Here we see why we need to consider the 4-complex J instead of the 4-graph J 4 alone: \(\delta _{3}(J_{4}) = 0\) because a 3-set abc ∉ E(H) has degree zero in J 4.

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Acknowledgements

The author would like to thank Jie Han for valuable discussion when preparing this manuscript. He also thanks Albert Bush, Jie Han, Allan Lo, Richard Mycroft, and Andrew Treglown for their comments that improved the presentation of the manuscript.

The author is partially supported by NSF grant DMS-1400073.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Georgia State University, Atlanta, GA, 30303, USA

    Yi Zhao

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Editors and Affiliations

  1. Dept of Math, Statistics & Comp.Sc, Macalester College, St. Paul, Minnesota, USA

    Andrew Beveridge

  2. Dept. of Mathematics, University of South Carolina, Columbia, South Carolina, USA

    Jerrold R. Griggs

  3. Dept. of Mathematics, Iowa State University, Ames, Iowa, USA

    Leslie Hogben

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Gregg Musiker

  5. School of Mathematics & Computer Sc, Georgia Institute of Technology, Atlanta, Georgia, USA

    Prasad Tetali

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Zhao, Y. (2016). Recent advances on Dirac-type problems for hypergraphs. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_6

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