Covering of graphs by complete bipartite subgraphs; Complexity of 0–1 matrices

Abstract

We prove that the edge set of an arbitrary simple graphG onn vertices can be covered by at mostn−[log2 n]+1 complete bipartite subgraphs ofG. If the weight of a subgraph is the number of its vertices, then there always exists a cover with total weightc(n 2/logn) and this bound is sharp apart from a constant factor. Our result answers a problem of T. G. Tarján.

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

References

  1. [1]

    J. C. Bermond, Couverture des arrêtes d’un graphe par des graphes bipartis complets,preprint, Univ. de Paris-Sud, Centre d’Orsay, Rapport de Recherche No. 10. (June 1978).

  2. [2]

    B. Bollobás, On complete subgraphs of different orders,Math. Proc. Comb. Philos. Soc. 79 (1976), 19–24.

    MATH  Article  Google Scholar 

  3. [3]

    N. G. DeBruijn andP. Erdős. On a combinatorial problem,Nederl. Akad. Wetensch. Proc. 51 (1948), 1277–1279.

    MathSciNet  Google Scholar 

  4. [4]

    F. R. K. Chung, On the coverings of graphs,Discrete Math. 30 (1980), 89–93.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    P. Erdős, Graph theory and probability,Canad. J. Math. 11 (1959) 34–38.

    MathSciNet  Google Scholar 

  6. [6]

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

    MathSciNet  Google Scholar 

  7. [7]

    P. Erdős andG. Szekeres, A combinatorial problem in geometry,Compositio Math. 2 (1935), 463–470.

    MathSciNet  Google Scholar 

  8. [8]

    R. L. Graham andL. Lovász, Distance matrix polynomials of trees,Advances in Mathematics 29 (1978), 60–88.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    E. Győri andA. V. Kostochka, On a problem of G. O. H. Katona and T. Tarján,Acta Math. Acad. Sci. Hungar. 34 (1979), 321–327.

    Article  MathSciNet  Google Scholar 

  10. [10]

    F. Harary, D. Hsu andZ. Miller, The biparticity of a graph,J. Graph Theory 1 (1977), 131–133.

    MATH  MathSciNet  Google Scholar 

  11. [11]

    C. Hylten-Cavallius, On a combinatorial problem,Colloq. Math. 6 (1958), 59–65.

    MATH  MathSciNet  Google Scholar 

  12. [12]

    G. Katona andE. Szemerédi, On a problem of graph theory,Studia Sci. Math. Hungar. 2 (1967), 23–28.

    MATH  MathSciNet  Google Scholar 

  13. [13]

    J. Lehel, Covers in hypergraphs,Combinatorica 2 (3) (1982), 305–309.

    MATH  MathSciNet  Google Scholar 

  14. [14]

    J. Lehel andZs. Tuza, Triangle-free partial graphs and edge covering theorems,Discrete Math. 39 (1982), 59–65.

    MATH  Article  MathSciNet  Google Scholar 

  15. [15]

    L. Lovász, On covering of graphs, in:Theory of Graphs, (P. Erdős and G. Katona, eds.),Proc. Coll. at Tihany, Hungary, 1966, Academic Press and Akadémiai Kiadó, 231–236.

  16. [16]

    V. Rödl, private communication.

  17. [17]

    T. G. Tarján, Complexity of lattice-configurations,Studia Sci. Math. Hungar. 10 (1975), 203–211.

    MathSciNet  Google Scholar 

  18. [18]

    H. Tverberg, On the decomposition ofK n in complete bipartite graphs,J. Graph Theory 6 (1982), 493–494.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Dedicated to Paul Erdős on his seventieth birthday

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Tuza, Z. Covering of graphs by complete bipartite subgraphs; Complexity of 0–1 matrices. Combinatorica 4, 111–116 (1984). https://doi.org/10.1007/BF02579163

Download citation

AMS subject classification (1980)

  • 05 C 35
  • 15 A 99