Distance-Preserving Subgraphs of Johnson Graphs

Abstract

We give a characterization of distance-preserving subgraphs of Johnson graphs, i.e., of graphs which are isometrically embeddable into Johnson graphs (the Johnson graph J(m,∧) has the subsets of cardinality m of a set ∧ as the vertex-set and two such sets A,B are adjacent iff |AΔB|=2). Our characterization is similar to the characterization of D. Ž. Djoković [11] of distance-preserving subgraphs of hypercubes and provides an explicit description of the wallspace (split system) defining the embedding.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    H.-J. Bandelt and V. Chepoi: Decomposition and l 1-embedding of weakly median graphs, Europ. J. Combin. 21 (2000), 701-714.

    Article  MATH  Google Scholar 

  2. [2]

    H.-J. Bandelt and V. Chepoi: Metric graph theory and geometry: a survey, in: J. E. Goodman, J. Pach, R. Pollack (Eds.), Surveys on Discrete and Computational Geometry. Twenty Years later, Contemp. Math., vol. 453, AMS, Providence, RI, 2008, 49-86.

    Google Scholar 

  3. [3]

    H.-J. Bandelt, V. Chepoi and K. Knauer: COMs: complexes of oriented matroids, arXiv:1507.06111, 2015.

    Google Scholar 

  4. [4]

    H.-J. Bandelt and A. W. M. Dress: A canonical decomposition theory for metrics on a nite set, Adv. Math. 92 (1992), 47-105.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    H.-J. Bandelt, V. Chepoi, A. Dress and J. Koolen: Combinatorics of lopsided sets, Europ. J. Combin. 27 (2006), 669-689.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler: Ori-ented Matroids, Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge, 1993.

    Google Scholar 

  7. [7]

    J. Chalopin, V. Chepoi and D. Osajda: On two conjectures of Maurer concerning basis graphs of matroids, J. Combin. Th. Ser. B 114 (2015), 1-32.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    V. Chepoi: d-Convexity and isometric subgraphs of Hamming graphs. Cybernetics 24 (1988), 6-10 (Russian, English transl.).

    MathSciNet  Article  Google Scholar 

  9. [9]

    M. Deza and M. Laurent: Geometry of Cuts and Metrics, Springer-Verlag, Berlin, 1997.

    Book  MATH  Google Scholar 

  10. [10]

    M. Deza and S. Shpectorov: Recognition of l1-graphs with complexity O(nm), or football in a hypercube, Europ. J. Combin. 17 (1996), 279-289.

    Article  MATH  Google Scholar 

  11. [11]

    D. Ž. Djokovic: Distance-preserving subgraphs of hypercubes, J. Combin. Th. Ser. B 14 (1973), 263-267.

    MathSciNet  Article  Google Scholar 

  12. [12]

    A. Dress, K. T. Huber, J. Koolen, V. Moulton and A. Spillner: Basic Phy-logenetic Combinatorics, Cambridge University Press, Cambridge, 2012.

    MATH  Google Scholar 

  13. [13]

    R. L. Graham and P. M. Winkler: On isometric embeddings of graphs, Trans. Amer. Math. Soc. 288 (1985), 527-536.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    F. Haglund and F. Paulin: Simplicité de groupes d’automorphismes d’espaces á courbure négative, The Epstein birthday schrift, Geom. Topol. Monogr. 1 (1998), 181-248 (electronic), Geom. Topol. Publ., Coventry.

    Article  MATH  Google Scholar 

  15. [15]

    W. Imrich and S. Klavžar: Product Graphs: Structure and Recognition, Wiley-Interscience Publication, New York, 2000.

    MATH  Google Scholar 

  16. [16]

    J. Lawrence: Lopsided sets and orthant-intersection of convex sets, Pacic J. Math. 104 (1983), 155-173.

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    S. B. Maurer: Matroid basis graphs I, J. Combin. Th. Ser. B 14 (1973), 216-240.

    MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    S. V. Shpectorov: On scale embeddings of graphs into hypercubes, Europ. J. Combin. 14 (1993), 117-130.

    MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    P. Terwilliger and M. Deza: The classification of finite connected hypermetric spaces, Graphs and Combin. 3 (1987), 293-298.

    MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    E. Wilkeit: Isometric embedding in Hamming graphs, J. Combin. Th. Ser. B 50 (1990), 179-197.

    MathSciNet  Article  MATH  Google Scholar 

  21. [21]

    P. M. Winkler: Isometric embedding in the product of complete graphs, Discr. Appl. Math. 7 (1984), 221-225.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Victor Chepoi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chepoi, V. Distance-Preserving Subgraphs of Johnson Graphs. Combinatorica 37, 1039–1055 (2017). https://doi.org/10.1007/s00493-016-3421-y

Download citation

Mathematics Subject Classification (2000)

  • 05C12
  • 52B40