Edge-Clique Covers of the Tensor Product

  • Wing-Kai Hon
  • Ton Kloks
  • Hsiang-Hsuan Liu
  • Yue-Li Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8546)

Abstract

In this paper we study the edge-clique cover number of the tensor product Kn ×Kn. We derive an easy lowerbound for the edge-clique number of graphs in general. We prove that, when n is prime θe(Kn ×Kn) matches the lowerbound. Moreover, we prove that θe(Kn ×Kn) matches the lowerbound if and only if a projective plane of order n exists. We also show an easy upperbound for θe(Kn ×Kn) in general, and give its limiting value when the Riemann hypothesis is true.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chang, M.-S., Müller, H.: On the tree-degree of graphs. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 44–54. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Cramér, H.: On the order of magnitude of the difference between consecutive prime numbers. Acta Arithmetica 2, 23–46 (1936)Google Scholar
  3. 3.
    Cygan, M., Pilipczuk, M., Pilipczuk, M.: Known algorithms for Edge Clique Cover are probably optimal. In: Proceedings of SODA 2013, pp. 1044–1053 (2013)Google Scholar
  4. 4.
    Bruijn, N.G., Erdős, D., On, P.: a combinatorial problem. Proceedings Koninklijke Nederlandse Akademie van Wetenschappen 51, 1277–1279 (1948)MATHGoogle Scholar
  5. 5.
    Erdős, P., Goodman, A., Pósa, L.: The representation of a graph by set intersections. Canadian Journal of Mathematics 18, 106–112 (1966)CrossRefGoogle Scholar
  6. 6.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Data reduction and exact algorithms for clique cover. ACM Journal of Experimental Algorithmics 13, article 2 (2009)Google Scholar
  7. 7.
    Gyárfás, A.: A simple lowerbound on edge covering by cliques. Discrete Mathematics 85, 103–104 (1990)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Hoover, D.N.: Complexity of graph covering problems for graphs of low degree. Journal of Combinatorial Mathematics and Combinatorial Computing 11, 187–208 (1992)MATHMathSciNetGoogle Scholar
  9. 9.
    Kong, J., Wu, Y.: On economical set representations of graphs. Discrete Mathematics and Theoretical Computer Science 11, 71–96 (2009)MATHMathSciNetGoogle Scholar
  10. 10.
    Kou, L.T., Stockmeyer, L.J., Wong, C.K.: Covering edges by cliques with regard to keyword conflicts and intersection graphs. Communications of the ACM 21, 135–139 (1978)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Moradi, S.: A note on tensor product of graphs. Iranian Journal of Mathematical Sciences and Informatics 7, 73–81 (2012)MathSciNetGoogle Scholar
  12. 12.
    Opsut, R.J.: On the computation of the competition number of a graph. SIAM Journal on Algebraic Discrete Methods 3, 420–428 (1982)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Orlin, J.B.: Contentment in graph theory: Covering graphs with cliques. Indagationes Mathematicae 80, 406–424 (1977)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Roberts, F.S.: Food webs, competition graphs, and the boxicity of ecological phase space. In: Theory and Applications of Graphs, pp. 477–490 (1978)Google Scholar
  15. 15.
    Roberts, F.S.: Applications of edge coverings by cliques. Discrete Applied Mathematics 10, 93–109 (1985)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Weisstein, E.: Bertrand’s postulate. Wolfram MathworldGoogle Scholar
  17. 17.
    Weisstein, E.: Projective plane. Wolfram MathworldGoogle Scholar
  18. 18.
    Cameron, P.J.: Projective and Polar Spaces. QMW Maths Notes 13 (1991)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Wing-Kai Hon
    • 1
  • Ton Kloks
    • 1
  • Hsiang-Hsuan Liu
    • 1
    • 3
  • Yue-Li Wang
    • 2
  1. 1.National Tsing Hua UniversityTaiwan
  2. 2.National Taiwan University of Science and TechnologyTaiwan
  3. 3.University of LiverpoolUK

Personalised recommendations