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Celoria, Daniele; Yerolemou, Naya

Filtered matchings and simplicial complexes. (English) Zbl 1496.05192

Australas. J. Comb. 82, Part 3, 335-352 (2022).
Summary: To any finite simplicial complex \(X\), we associate a natural filtration starting from M. K. Chari and M. Joswig’s [Discrete Math. 302, No. 1–3, 39–51 (2005; Zbl 1091.57025)] discrete Morse complex and abutting to the matching complex of \(X\). This construction leads to the definition of several homology theories, which we compute in a number of examples. We also completely determine the graded object associated to this filtration in terms of the homology of simpler complexes. This last result provides some connections to the number of vertex-disjoint cycles of a graph.

Cited in 3 Documents

MSC:

05E45 Combinatorial aspects of simplicial complexes
05C30 Enumeration in graph theory
05C38 Paths and cycles

Keywords:

vertex-disjoint cycles of a graph; homology of simple complexes

Citations:

Zbl 1091.57025

Software:

SageMath
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Full Text: arXiv Link

References:

[1] N. Biggs,Algebraic graph theory, Cambridge University Press, vol. 67 (1993). · Zbl 0797.05032
[2] N. A. Capitelli and E. G. Minian, A simplicial complex is uniquely determined by its set of discrete Morse functions,Discrete Comput. Geom.58(1) (2017), 144-157. · Zbl 1371.05326
[3] M. K. Chari and M. Joswig, Complexes of discrete Morse functions,Discrete Math.302(1-3) (2005), 39-51. · Zbl 1091.57025
[4] G. Dirac and P. Erd˝os, On the maximal number of independent circuits in a graph,Acta Math. Hungar.14(1-2) (1963), 79-94. · Zbl 0122.24903
[5] Y. Egawa, M. Hagita, K. Kawarabayashi and H. Wang, Covering vertices of a graph by k disjoint cycles,Discrete Math.270(1-3) (2003), 115-125. · Zbl 1030.05064
[6] R. Forman, Morse theory for cell complexes,Adv. Math.134 (1998), 90-145. · Zbl 0896.57023
[7] J. Jonsson,Simplicial complexes of graphs, Springer Science & Business Media, (2007). · Zbl 1152.05001
[8] J. Jonsson, More torsion in the homology of the matching complex,Exp. Math. 19(3) (2010), 363-383. · Zbl 1247.05271
[9] M. Khovanov, A categorification of the Jones polynomial,Duke Math. J.101(3) (2000), 359-426. · Zbl 0960.57005
[10] D. N. Kozlov, Complexes of directed trees,J. Combin. Theory Ser. A88(1) (1999), 112-122. · Zbl 0934.05041
[11] M. Lin and N. A. Scoville, On the automorphism group of the Morse complex, Adv. Appl. Math.131 (2021), 102250. · Zbl 1475.57038
[12] M. Marietti and D. Testa, A uniform approach to complexes arising from forests, Electron. J. Combin.(2008), #R101. · Zbl 1178.05103
[13] The Sage Developers,SageMath, the Sage Mathematics Software System (Version 9.0), (2020),https://www.sagemath.org.
[14] J. Shareshian and M. L. Wachs, Torsion in the matching complex and chessboard complex,Adv. Math.212(2) (2007), 525-570 · Zbl 1117.05110
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