Skip to main content
SpringerLink
Log in

Homotopy and Hom Construction in the Category of Finite Hypergraphs

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

We define notions of a weak homotopy for finite hypergraphs and an exponential hypergraph with a right adjoint to the categorical product of finite hypergraphs, and investigate a connection between the weak homotopy and the exponential hypergraph. Then we discuss a Hom construction associated to a pair of finite hypergraphs and prove that the homotopy of hypergraph homomorphisms, defined in (Grigor’Yan et al. in Topol Appl 267:106877, 2019), could be characterized by properties of the Hom construction. In addition, we establish some properties of Hom constructions involving the categorical product of finite hypergraphs. As an application we show that the homotopy of d-colorings of a simple hypergraph \({\mathcal {H}}\) could be characterized by properties of Hom constructions associated to the maximum simple hypergraph and \({\mathcal {H}}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Alexandroff, P.S.: Diskrete Räume. Mathematicheskii Sbornik(N.S.) 2, 501–518 (1937)

  2. Alon, N., Frankl, P., Lovász, L.: The chromatic number of Kneser hypergraphs. Trans. Am. Math. Soc. 298(1), 359–370 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Björner, A.: Topological Methods. In: Handbook of Combinatorics vol 1. pp. 1819–1872. Elsevier, Amsterdam. (1995)

  4. Babson, E., Kozlov, D.N.: Complexes of graph homomorphisms. Israel J. Math. 152, 285–312 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Babson, E., Kozlov, D.N.: Proof of the Lovász conjecture. Ann. Math. 165(3), 965–1007 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bar-Noy, A., Cheilaris, P., Olonetsky, S., Smorodinsky, S.: Online Conflict-Free Colorings for Hypergraphs. In: Arge L., Cachin C., Jurdziński T., Tarlecki A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73420-8_21

  7. Barmak, J.A.: Algebraic topology of finite topological spaces and applications. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  8. Breuer, F., Dall, A., Kubitzke, M.: Hypergraph coloring complexes. Discret. Math. 312(16), 2407–2420 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bressan, S., Li, J., Ren, S., Wu, J.: The embedded homology of hypergraphs and applications. Asian J Math 23(3), 1–18 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Caro, Y., Tuza, Z.: Hypergraph coverings and local colorings. J Comb Theory Ser B 52(1), 79–85 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dörfler, W., Waller, D.A.: A category-theoretical approach to hypergraphs. Arch. Math. 34(1), 185–192 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dochtermann, A.: Hom complexes and homotopy theory in the category of graphs. Eur. J. Comb. 30, 490–509 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dochtermann, A.: Homotopy groups of Hom complexes of graphs. J Comb Theory Ser A 116(1), 180–194 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Francisco, C.A., Hà, H.T., Tuyl, A.V.: Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals. J. Algebra 331(1), 224–242 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghosh, P., Pal, A.: Coloring of hypergraphs. J Math Inf 8, 37–44 (2017)

    Article  Google Scholar 

  16. Grigor’Yan, A., Jimenez, R., Muranov, Y., Yau, S.T.: Homology of path complexes and hypergraphs. Topol Appl 267, 106877 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hellmuth, M., Ostermeier, L., Stadler, P.F.: A survey on hypergraph products. Math. Comput. Sci. 6, 1–32 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Iriye, K., Kishimoto, D.: Hom complexes and hypergraph colorings. Topol Appl 160(12), 1333–1344 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jonsson, J.: Simplicial Complexes of Graphs. Lecture Notes in Mathematics, vol. 192. Springer-Verlag, Berlin (2008)

  20. Kozlov, D.N.: Chromatic numbers, morphism complexes, and Stiefel-Whitney characteristic classes (2005). https://doi.org/10.48550/arXiv:math/0505563

  21. Kozlov, D.N.: Simple homotopy types of Hom-complexes, neighborhood complexes, Lovász complexes, and atom crosscut complexes. Topol Appl. 153(14), 2445–2454 (2006)

    Article  MATH  Google Scholar 

  22. Kozlov, D.N.: Combinatorial algebraic topology. Algorithms and computation in mathematics, vol. 21. Springer, Berlin (2008)

    Google Scholar 

  23. Lange, C.: On generalised Kneser colourings. Mathematics (2003)

  24. Mirzakhani, M.,Vondrák, J.: Sperner’s Colorings, Hypergraph Labeling Problems and Fair Division. Proceedings of the 2015 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2015, 873-886 (2015)

  25. Matsushita, T.: Morphism complexes of sets with relations. Osaka J. Math. 53, 267–283 (2016)

    MathSciNet  MATH  Google Scholar 

  26. Matsushita, T.: Box complexes and homotopy theory of graphs. Homol Homot Appl 19(2), 175–197 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Matsushita, T.: Homotopy types of the Hom complexes of graphs. Eur. J. Comb. 63(2017), 216–226 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Singh, A.: Hom complexes of graphs of diameter 1(2019). https://doi.org/10.48550 / arXiv:1807.10498

  29. Thansri, T.: simple \(\Sigma _r\)-homotopy types of Hom complexes and box complexes assigned to \(r\)-graphs. Kyushu J Math 2012(66), 493–508 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This research was supported by the National Natural Science Foundations of China (No. 11771116) and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2022L266).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China

    Conglei Zhang & Yanying Wang

  2. College of Arts and Sciences, Shanxi Agricultural University, Jinzhong, 030801, People’s Republic of China

    Conglei Zhang

  3. School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan, 030603, People’s Republic of China

    Zhiguo Zhang

  4. College of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, 050061, People’s Republic of China

    Wei Dai

Authors
  1. Conglei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yanying Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhiguo Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wei Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Yanying Wang.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, C., Wang, Y., Zhang, Z. et al. Homotopy and Hom Construction in the Category of Finite Hypergraphs. Graphs and Combinatorics 39, 78 (2023). https://doi.org/10.1007/s00373-023-02672-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00373-023-02672-6

Keywords

Mathematics Subject Classification

Search

Navigation