Advertisement

Graphs and Combinatorics

, Volume 24, Issue 4, pp 291–301 | Cite as

Hadwiger Number and the Cartesian Product of Graphs

  • L. Sunil Chandran
  • Alexandr KostochkaEmail author
  • J. Krishnam Raju
Article

Abstract

The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph K n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), where χ(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product \(G \square H\) of graphs.

As the main result of this paper, we prove that \(\eta (G_1 \square G_2) \ge h\sqrt{l}\left (1 - o(1) \right )\) for any two graphs G 1 and G 2 with η(G 1) = h and η(G 2) = l. We show that the above lower bound is asymptotically best possible when h ≥ l. This asymptotically settles a question of Z. Miller (1978).

As consequences of our main result, we show the following:
  1. 1.

    Let G be a connected graph. Let \(G = G_1 \square G_2 \square ... \square G_k\) be the (unique) prime factorization of G. Then G satisfies Hadwiger’s conjecture if k ≥ 2 log log χ(G) + c′, where c′ is a constant. This improves the 2 log χ(G) + 3 bound in [2].

     
  2. 2.

    Let G 1 and G 2 be two graphs such that χ(G 1) ≥ χ(G 2) ≥ c log1.5(χ(G 1)), where c is a constant. Then \(G_1 \square G_2\) satisfies Hadwiger’s conjecture.

     
  3. 3.

    Hadwiger’s conjecture is true for G d (Cartesian product of G taken d times) for every graph G and every d ≥ 2. This settles a question by Chandran and Sivadasan [2]. (They had shown that the Hadiwger’s conjecture is true for G d if d ≥ 3).

     

Keywords

Hadwiger Number Hadwiger’s Conjecture Graph Cartesian product Minor Chromatic number 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aurenhammer, F., Hagauer, J., Imrich, W.: Cartesian graph factorization at logarithmic cost per edge. Computational Complexity 2, 331–349 (1992)Google Scholar
  2. 2.
    Chandran, L.S., Sivadasan, N.: On the Hardwiger’s conjecture for graph products. Discrete Mathematics 307, 266–273 (2007)Google Scholar
  3. 3.
    Diestel, R.: Graph Theory. Springer Verlag, New York (2000)Google Scholar
  4. 4.
    Diestel, R., Rempel, C.: Dense minors in graphs of large girth. Combinatorica 25, 111–116 (2005)Google Scholar
  5. 5.
    Dirac, G.A.: In abstrakten Graphen vorhandene vollständige 4–Graphen und ihre Unterteilungen. Math. Nachr. 22, 61–85 (1960)Google Scholar
  6. 6.
    Hadwiger, H.: Über eine Klassifikation der Streckenkomplexe, Vierteljscr. Naturforsch. Gessellsch. Zürich 88, 132–142 (1943)Google Scholar
  7. 7.
    Imrich, W., Klavžar, S.: Product Graphs: Structure and Recognition. Wiley, New York (2000)Google Scholar
  8. 8.
    Ivančo, J.: Some results on the Hadwiger number of graphs. Math. Slovaca 38, 221–226 (1988)Google Scholar
  9. 9.
    Iwaniec, H., Pintz, J.: Primes in short intervals. Monatsh. Math. 98, 115–143 (1984)Google Scholar
  10. 10.
    Kostochka, A.V.: The minimum hadwiger number of graphs with a given mean degree of vertices, Metody Diskret. Analiz. 38, 37–58 (1982) (In Russian)Google Scholar
  11. 11.
    Kotlov, A.: Minors and strong products. European Journal of Combinatorics 22, 511–512 (2001)Google Scholar
  12. 12.
    Kühn, D., Osthus, D.: Minors in graphs of large girth. Random Structures and Algorithms 22, 213–225 (2003)Google Scholar
  13. 13.
    Mader, W.: Homomorphiesätze für Graphen. Math. Annalen 178, 154–168 (1968)Google Scholar
  14. 14.
    Miller, Z.: Contractions of graphs: A theorem of Ore and an extremal problem. Discrete Mathematics 21, 261–273 (1978)Google Scholar
  15. 15.
    Robertson, N., Seymour, P.D., Thomas, R.: Hadwiger’s conjecture for K6-free graphs. Combinatorica 13, 279–361 (1993)Google Scholar
  16. 16.
    Ryser, H.J.: Combinatorial mathematics, The Carus Mathematical Monographs. 14 Wiley, New York (1963)Google Scholar
  17. 17.
    Sabidussi, G.: Graphs with given group and given graph theoretic properties. Canad. J. Math. 9, 515–525 (1957)Google Scholar
  18. 18.
    Thomason, A.G.: An extremal function for contractions of graphs. Math. Proc. Camb. Phil. Soc. 95, 261–265 (1984)Google Scholar
  19. 19.
    Wagner, K.: Über eine Eigenschaft der Ebenen Komplexe. Math. Annalen 114, 570–590 (1937)Google Scholar
  20. 20.
    Wagner, K.: Beweis einer Abschwächung der Hadwiger–Vermutung. Math. Annalen 153, 139–141 (1964)Google Scholar
  21. 21.
    West, D.B.: Introduction to Graph Theory. Prentice Hall India, NewDelhi (2003)Google Scholar

Copyright information

© Springer Japan 2008

Authors and Affiliations

  • L. Sunil Chandran
    • 1
  • Alexandr Kostochka
    • 2
    • 3
    Email author
  • J. Krishnam Raju
    • 4
  1. 1.Department of Computer Science and AutomationIndian Institute of ScienceBangaloreIndia
  2. 2.University of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Institute of MathematicsNovosibirskRussia
  4. 4.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations