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Matching Complexes of Trees and Applications of the Matching Tree Algorithm

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Abstract

A matching complex of a simple graph G is a simplicial complex with faces given by the matchings of G. The topology of matching complexes is mysterious; there are few graphs for which the homotopy type is known. Marietti and Testa showed that matching complexes of forests are contractible or homotopy equivalent to a wedge of spheres. We study two specific families of trees. For caterpillar graphs, we give explicit formulas for the number of spheres in each dimension and for perfect binary trees we find a strict connectivity bound. We also use a tool from discrete Morse theory called the Matching Tree Algorithm to study the connectivity of honeycomb graphs, partially answering a question raised by Jonsson.

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Notes

  1. For this configuration, the choice of splitting vertex does not affect the size or quantity of critical cells, but this is not true for all configurations.

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Acknowledgements

This work was partially completed during the 2019 Graduate Research Workshop in Combinatorics. The workshop was partially funded by NSF grants 1603823, 1604773 and 1604458, “Collaborative Research: Rocky Mountain— Great Plains Graduate Research Workshops in Combinatorics,” NSA grant H98230-18-1-0017, “The 2018 and 2019 Rocky Mountain—Great Plains Graduate Research Workshops in Combinatorics,” Simons Foundation Collaboration Grants #316262 and #426971 and grants from the Combinatorics Foundation and the Institute for Mathematics and its Applications. Additional funding was provided by Grant #174034 of the Ministry of Education, Science and Technological Development of Serbia. We would like to thank Margaret Bayer and Bennet Goeckner for their insights and guidance on this project. Additionally, we would like to thank Benjamin Braun, Russ Woodroofe, Mario Marietti, and Damiano Testa for their helpful comments on this manuscript. Finally, we would like to thank the anonymous referee whose comments sketched a proof Theorem 5.19.

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Correspondence to Marija Jelić Milutinović.

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Communicated by Bridget Tenner.

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Appendix A. Computations for the Homotopy Type of Trees

Appendix A. Computations for the Homotopy Type of Trees

Table 1 The number of spheres of dimension d in the homotopy type of \(M(G_{n}(m_1, \ldots , m_n))\), where \(G_{n}(m_1, \ldots , m_n)\) is a caterpillar with \(m_i=t_i+1\) legs at each vertex i of the central path

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Jelić Milutinović, M., Jenne, H., McDonough, A. et al. Matching Complexes of Trees and Applications of the Matching Tree Algorithm. Ann. Comb. 26, 1041–1075 (2022). https://doi.org/10.1007/s00026-022-00605-3

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