Thep-intersection number of a complete bipartite graph and orthogonal double coverings of a clique


Thep-intersection graph of a collection of finite sets {S i } n i=1 is the graph with vertices 1, ...,n such thati, j are adjacent if and only if |S i S j |≥p. Thep-intersection number of a graphG, herein denoted θ p (G), is the minimum size of a setU such thatG is thep-intersection graph of subsets ofU. IfG is the complete bipartite graphK n,n andp≥2, then θ p (K n, n )≥(n 2+(2p−1)n)/p. Whenp=2, equality holds if and only ifK n has anorthogonal double covering, which is a collection ofn subgraphs ofK n , each withn−1 edges and maximum degree 2, such that each pair of subgraphs shares exactly one edge. By construction,K n has a simple explicit orthogonal double covering whenn is congruent modulo 12 to one of {1, 2, 5, 7, 10, 11}.

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


  1. [1]

    B. Alspach, K. Heinrich, andG. Liu: Orthogonal factorizations of graphs, in:Contemporary Design Theory: A collection of Surveys (J. Dinitz and D. Stinson, eds.), (Wiley 1992), 13–40.

  2. [2]

    F. E. Bennet, andL. Wu: On minimum matrix representation of closure operations,Discrete Appl. Math. 26 (1990), 25–40.

    Google Scholar 

  3. [3]

    R. C. Brigham, R. D. Dutton, andF. R. McMorris: On the relationship betweenp-edge andp-vertex clique covers,Vishwa Intl. J. Graph Theory 1 (1992), 133–140.

    Google Scholar 

  4. [4]

    J. Demetrovics, Z. Füredi andG. O. H. Katona: Minimum matrix representation of closure operations,Discrete Appl. Math. 11 (1985), 115–128.

    Google Scholar 

  5. [5]

    N. Eaton, R. J. Gould, andV. Rödl: Onp-intersection representations, preprint.

  6. [6]

    P. Erdős, A. Goodman, andL. Pósa: The representation of a graph by set intersections,Canad. J. Math. 18 (1966), 106–112.

    Google Scholar 

  7. [7]

    P. Frankl, andV. Rödl: Near perfect coverings in graphs and hypergraphs,Europ. J. Comb. 6 (1985), 317–326.

    Google Scholar 

  8. [8]

    Z. Füredi: Matchings and covers in hypergraphs,Graphs and Combinatorics 4 (1988), 115–206.

    Google Scholar 

  9. [9]

    Z. Füredi: A random representation of the complete bipartite graphs, submitted.

  10. [10]

    B. Ganter, andH.-D. O. F. Gronau: On two conjectures of Demetrovics, Füredi, and Katona on partitions,Discrete Math. 88 (1991), 149–155.

    Google Scholar 

  11. [11]

    B. Ganter, H.-D. O. F. Gronau, andR. C. Mullin: On orthogonal double covers ofK n ,Ars Combinatoria 37 (1994), 209–221.

    Google Scholar 

  12. [12]

    M. C. Golumbic, C. L. Monna, andW. T. Trotter: Tolerance graphs,Disc. Appl. Math. 9 (1984), 157–170.

    Google Scholar 

  13. [13]

    K. Heinrich, andG. Nonay: Path and cycle decompositions of complete multigraphs,Annals Discr. Math. 27 (1985), 275–286.

    Google Scholar 

  14. [14]

    F. Hering: Block designs with cyclic block structure,Annals Discr. Math. 6 (1980), 201–214.

    Google Scholar 

  15. [15]

    G. Isaak, S.-R. Kim, T. A. McKee, F. R. McMorris, andF. S. Roberts: 2-competition graphs,SIAM J. Discr. Math. 5 (1992), 524–538.

    Google Scholar 

  16. [16]

    M. S. Jacobson: On thep-edge clique cover number of complete bipartite graphs,SIAM J. Discr. Math. 5 (1992), 539–544.

    Google Scholar 

  17. [17]

    M. S. Jacobson, F. R. McMorris, andH. M. Mulder: Tolerance intersection graphs, in:Graph Theory, Combinatorics, and Applications, Vol 2 (Proc. 6th Intl. Conf. Kalamazoo, 1988). Y. Alavi et al, eds. (Wiley 1991), 705–723.

  18. [18]

    M. S. Jacobson, F. R. McMorris, andE. R. Scheinerman: General results on tolerance intersection graphs,J. Graph Theory,15 (1991), 573–577.

    Google Scholar 

  19. [19]

    S.-R. Kim: Competition graphs and scientific laws for food webs and other systems, Ph. D. Thesis, Rutgers Univ., 1988.

  20. [20]

    S.-R. Kim, T. A. McKee, F. R. McMorris, andF. S. Roberts:p-competition graphs, DIMACS Technical Report 89-19, see also:p-competition numbers,Discrete Appl. Math. 46 (1993), 87–92;p-competitions graphs,Linear Alg. Appl. (to appear).

    Google Scholar 

  21. [21]

    G. E. Major:p-edge clique coverings of graphs, M. A. Thesis, Univ. of Louisville, 1990.

  22. [22]

    G. E. Major, andF. R. McMorris:p-edge clique coverings of graphs,Congressus Num. 79 (1990), 143–145.

    Google Scholar 

  23. [23]

    A. Rausche: On the existence of special block designs.Rostock. Math. Kolloq 35 (1988), 13–20.

    Google Scholar 

Download references

Author information



Additional information

Research supported in part by ONR Grant N00014-5K0570.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chung, M.S., West, D.B. Thep-intersection number of a complete bipartite graph and orthogonal double coverings of a clique. Combinatorica 14, 453–461 (1994).

Download citation

AMS subject classification code (1991)

  • 05 C 35
  • 05 C 70
  • 90 C 90