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


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).



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


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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

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