Lower bounds for boxicity

Abstract

An axis-parallel b-dimensional box is a Cartesian product R 1×–2×...×R b where R i is a closed interval of the form [a i ; b i ] on the real line. For a graph G, its boxicity box(G) is the minimum dimension b, such that G is representable as the intersection graph of boxes in b-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which are listed below:

  1. 1.

    The boxicity of a graph on n vertices with no universal vertices and minimum degree δ is at least n/2(nδ−1).

  2. 2.

    Consider the G(n;p) model of random graphs. Let p ≤ 1 − 40logn/n 2. Then with high probability, box(G) = Ω(np(1 − p)). On setting p = 1/2 we immediately infer that almost all graphs have boxicity Ω(n). Another consequence of this result is as follows: For any positive constant c < 1, almost all graphs on n vertices and \(m \leqslant c\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\) edges have boxicity Ω(m/n).

  3. 3.

    Let G be a connected k-regular graph on n vertices. Let λ be the second largest eigenvalue in absolute value of the adjacency matrix of G. Then, the boxicity of G is at least \(\left( {\frac{{k^2 /\lambda ^2 }} {{\log \left( {1 + k^2 /\lambda ^2 } \right)}}} \right)\left( {\frac{{n - k - 1}} {{2n}}} \right)\).

  4. 4.

    For any positive constant c < 1, almost all balanced bipartite graphs on 2n vertices and mcn 2 edges have boxicity Ω(m/n).

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References

  1. [1]

    A. Adiga, J. Babu and L. S. Chandran: A constant factor approximation algorithm for boxicity of circular arc graphs, in: F. Dehne, J. Iacono, J.-R. Sack (eds.), WADS, vol. 6844 of Lecture Notes in Computer Science, Springer, 2011.

    Google Scholar 

  2. [2]

    N. Alon: Eigenvalues and expanders, Combinatorica 6 (1986), 83–96.

    Article  MATH  MathSciNet  Google Scholar 

  3. [3]

    N. Alon, J. H. Spencer and P. Erdös: The probabilistic method, John Wiley & Sons, Inc., 1992.

    MATH  Google Scholar 

  4. [4]

    S. Bellantoni, I. B.-A. Hartman, T. Przytycka and S. Whitesides: Grid intersection graphs and boxicity, Disc. Math. 114) (1993), 41–49.

    Article  MATH  MathSciNet  Google Scholar 

  5. [5]

    A. Bielecki: Problem 56, Colloq. Math 1 (1948), 333.

    Google Scholar 

  6. [6]

    N. Biggs: Algebraic Graph theory, Cambridge University Press Cambridge, 1993.

    Google Scholar 

  7. [7]

    Y. Bilu and N. Linial: Monotone maps, sphericity and bounded second eigenvalue, J. Comb. Theory Ser. B 95 (2005), 283–299.

    Article  MATH  MathSciNet  Google Scholar 

  8. [8]

    B. Bollobás: Random graphs, Cambridge university press, 2001.

    Book  MATH  Google Scholar 

  9. [9]

    B. Bollobás: Combinatorics, Cambridge University Press, 1986.

    MATH  Google Scholar 

  10. [10]

    L. S. Chandran, M. C. Francis and N. Sivadasan: Boxicity and maximum degree, J. Comb. Theory Ser. B 98 (2008), 443–445.

    Article  MATH  MathSciNet  Google Scholar 

  11. [11]

    L. S. Chandran and N. Sivadasan: Boxicity and treewidth, J. Comb. Theory Ser. B 97 (2007), 733–744.

    Article  MATH  MathSciNet  Google Scholar 

  12. [12]

    Y. W. Chang and D. B. West: Interval number and boxicity of digraphs, in: Proceedings of the 8th International Graph Theory Conf., 1998.

    Google Scholar 

  13. [13]

    Y. W. Chang and D. B. West: Rectangle number for hyper cubes and complete multipartite graphs, in: 29th SE conf. Comb., Graph Th. and Comp., Congr. Numer. 132, 1998.

  14. [14]

    M. B. Cozzens: Higher and multi-dimensional analogues of interval graphs, Ph.D. thesis, Department of Mathematics, Rutgers University New Brunswick, NJ (1981).

    Google Scholar 

  15. [15]

    M. B. Cozzens and F. S. Roberts: Computing the boxicity of a graph by covering its complement by cointerval graphs, Disc. Appl. Math. 6 (1983), 217–228.

    Article  MATH  MathSciNet  Google Scholar 

  16. [16]

    R. B. Feinberg: The circular dimension of a graph, Disc. Math. 25 (1979), 27–31.

    Article  MATH  MathSciNet  Google Scholar 

  17. [17]

    P. C. Fishburn: On the sphericity and cubicity of graphs, J. Comb. Theory Ser. B 35 (1983), 309–308.

    Article  MATH  MathSciNet  Google Scholar 

  18. [18]

    R. J. Fowler, M. S. Paterson and S. L. Tanimoto: Optimal packing and covering in the plane are NP-complete, Information Processing letters 12 (1981), 133–137.

    Article  MATH  MathSciNet  Google Scholar 

  19. [19]

    J. Friedman: A proof of Alon’s second eigenvalue conjecture, accepted to the Memoirs of the A.M.S.

  20. [20]

    L. H. Harper: Global methods for combinatorial isoperimetric problems, Cambridge University Press Cambridge, 2004.

    Book  MATH  Google Scholar 

  21. [21]

    T. F. Havel: The combinatorial distance geometry approach to the calculation of molecular conformation, Ph.D. thesis, University of California Berkeley (1982).

    Google Scholar 

  22. [22]

    H. Imai and T. Asano: Finding the connected component and a maximum clique of an intersection graph of rectangles in the plane, Journal of algorithms 4 (1983), 310–323.

    Article  MATH  MathSciNet  Google Scholar 

  23. [23]

    N. Kahale: Expander codes, Ph.D. thesis, MIT (1993).

    Google Scholar 

  24. [24]

    A. Kostochka: Coloring intersection graphs of geometric figures with a given clique number, Contemporary mathematics 342 (2004), 127–138.

    Article  MathSciNet  Google Scholar 

  25. [25]

    J. Kratochvíl: A special planar satisfiability problem and a consequence of its NP-completeness, Disc. Appl. Math. 52 (1994), 233–252.

    Article  MATH  Google Scholar 

  26. [26]

    H. Maehara, J. Reiterman, V. Rödl and E. Šiňajová: Embedding of trees in euclidean spaces, Graphs and combinatorics 4 (1988), 43–47.

    Article  MATH  MathSciNet  Google Scholar 

  27. [27]

    T. A. McKee and E. R. Scheinerman: On the chordality of a graph, Journal of Graph Theory 17 (1993), 221–232.

    Article  MATH  MathSciNet  Google Scholar 

  28. [28]

    M. Mitzenmacher and E. Upfal: Probability and computing, Cambridge University Press Cambridge, 2005.

    Book  MATH  Google Scholar 

  29. [29]

    F. S. Roberts: Recent Progresses in Combinatorics, chap. On the boxicity and cubicity of a graph, Academic Press New York, 1969, 301–310.

    Google Scholar 

  30. [30]

    E. R. cheinerman: Intersection classes and multiple intersection parameters, Ph.D. thesis, Princeton University (1984).

    Google Scholar 

  31. [31]

    J. B. Shearer: A note on circular dimension, Disc. Math. 29 (1980), 103–103.

    Article  MATH  MathSciNet  Google Scholar 

  32. [32]

    R. M. Tanner: Explicit construction of concentrators from generalized n-gons, SIAM J. Algebraic discrete methods 5 (1984), 287–294.

    Article  MATH  MathSciNet  Google Scholar 

  33. [33]

    C. Thomassen: Interval representations of planar graphs, J. Comb. Theory Ser. B 40 (1986), 9–20.

    Article  MATH  MathSciNet  Google Scholar 

  34. [34]

    W. T. Trotter, Jr.: A forbidden subgraph characterization of Roberts’ inequality for boxicity, Disc. Math. 28 (1979), 303–314.

    Article  MATH  MathSciNet  Google Scholar 

  35. [35]

    W. T. Trotter, Jr. and D. B. West: Poset boxicity of graphs, Disc. Math. 64 (1987), 105–107.

    Article  MATH  MathSciNet  Google Scholar 

  36. [36]

    N. C. Wormald: Surveys in combinatorics, 1999 (Canterbury), chap. Models of random graphs, Cambridge Univ. Press Cambridge, 1999, 239–298.

    Google Scholar 

  37. [37]

    B. D. McKay and N. C. Wormald: Asymptotic Enumeration By Degree Sequence of Graphs With Degrees o(n 1/2), Combinatorica 11 (1991), 369–382.

    Article  MATH  MathSciNet  Google Scholar 

  38. [38]

    M. Yannakakis: The complexity of the partial order dimension problem, SIAM J. Alg. Disc. Math. 3 (1982), 351–358.

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Abhijin Adiga or L. Sunil Chandran or Naveen Sivadasan.

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Adiga, A., Chandran, L.S. & Sivadasan, N. Lower bounds for boxicity. Combinatorica 34, 631–655 (2014). https://doi.org/10.1007/s00493-011-2981-0

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Mathematics Subject Classification (2000)

  • 05C62
  • 05C80