Graph Bases and Diagram Commutativity

Original Paper


Given two cycles A and B in a graph, such that \(A\cap B\) is a non-trivial path, the connected sum \(A\hat{+} B\) is the cycle whose edges are the symmetric difference of E(A) and E(B). A special kind of cycle basis for a graph, a connected sum basis, is defined. Such a basis has the property that a hierarchical method, building successive cycles through connected sum, eventually reaches all the cycles of the graph. It is proved that every graph has a connected sum basis. A property is said to be cooperative if it holds for the connected sum of two cycles when it holds for the summands. Cooperative properties that hold for the cycles of a connected sum basis will hold for all cycles in the graph. As an application, commutativity of a groupoid diagram follows from commutativity of a connected sum basis for the underlying graph of the diagram. An example is given of a noncommutative diagram with a (non-connected sum) basis of cycles which do commute.


Cycle basis Connected sum Commutative diagram Groupoid Robust cycle basis Ear basis Geodesic cycle 

Mathematics Subject Classification

05C38 20L05 18A10 



We thank the referees for helpful comments.


  1. 1.
    Diestel, R.: Graph Theory, Graduate Texts in Mathematics, vol. 173, 3rd edn. Springer, Berlin (2005)Google Scholar
  2. 2.
    Dixon, E.T., Goodman, S.E.: An algorithm for the longest cycle problem. Networks 6, 139–146 (1976)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
  4. 4.
    Galluccio, A., Loebl, M.: \((p, q)\)-odd digraphs. J. Graph Theory 23(2), 175–184 (1996)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hammack, R.H., Kainen, P.C.: Robust cycle bases do not exist for \(K_{n, n}\) if \(n \ge 8\). Discrete Appl. Math. 235, 206–211 (2018)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)CrossRefMATHGoogle Scholar
  7. 7.
    Kainen, P.C.: On robust cycle bases. Electron. Notes Discret. Math. 11, 430–437 (2002) [Proc. 9th Quadr. Conf. on Graph Theory, Comb., Algorithms and Appl., ed. by Y. Alavi et al., 2000)]Google Scholar
  8. 8.
    Kainen, P.C.: Isolated squares in hypercubes and robustness of commutativity. Cahiers de topologie et géom. diff. catég 43(3), 213–220 (2002)MathSciNetMATHGoogle Scholar
  9. 9.
    Kainen, P.C.: Graph cycles and diagram commutativity. Diagrammes 67–68, 177–238 (2012) (Supplem.) Google Scholar
  10. 10.
    Kainen, P.C.: Cycle construction and geodesic cycles with application to the hypercube. Ars Math. Contemp. 9(1), 27–43 (2015)MathSciNetMATHGoogle Scholar
  11. 11.
    Klemm, K., Stadler, P.F.: Statistics of cycles in large networks. Phys. Rev. E 73, 025101(R) (2006)CrossRefGoogle Scholar
  12. 12.
    Klemm, K., Stadler, P.F.: A note on fundamental, non-fundamental, and robust cycle bases. Discrete Appl. Math. 157, 2432–2438 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lin, Y.J.: Connected sum construction of constant \(Q\)-curvature maniolds in higher dimensions. Differ. Geom. Appl. 40, 290–320 (2015)CrossRefMATHGoogle Scholar
  14. 14.
    Loebl, M., Matamala, M.: Some remarks on cycles in graphs and digraphs. Discret. Math. 233(1–3), 175–182 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn., Graduate Texts in Mathematics (Book 5). Springer, New York (1998)Google Scholar
  16. 16.
    Milnor, J.: A unique decomposition theorem for 3-manifolds. Am. J. Math. 84, 1–7 (1962)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    OEIS Foundation Inc.: The On-Line encyclopedia of integer sequences. (2018)
  18. 18.
    Ostermeier, P.-J., Hellmuth, M., Klemm, K., Leydold, J., Stadler, P.F.: A note on quasi-robust cycle bases. Ars Math. Contemp. 2, 231–240 (2009)MathSciNetMATHGoogle Scholar
  19. 19.
    Spanier, E.H.: Algebraic Topology. McGraw-Hill, New York (1966)MATHGoogle Scholar
  20. 20.
    Wall, C.T.C.: Classification problems in differential topology, V. Invent. Math. 1, 355–374 (1966) [corrigendum, ibid 2, 306 (1966)]Google Scholar
  21. 21.
    White, A.T.: Graphs of Groups on Surfaces. Elsevier, Amsterdam (2001)MATHGoogle Scholar
  22. 22.
    Whitney, H.: Non-separable and planar graphs. Trans. AMS 34, 339–362 (1932)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Box 2014Virginia Commonwealth UniversityRichmondUSA
  2. 2.Department of Mathematics and StatisticsGeorgetown UniversityWashingtonUSA

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