Chandran, L Sunil and Kostochka, Alexandr and Raju, J Krishnam (2008) Hadwiger Number and the Cartesian Product of Graphs. In: Graphs and Combinatorics, 24 (4). pp. 291-301.
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The Hadwiger number eta(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, eta(G) >= chi(G), where chi(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)) >= h root 1 (1 - o(1)) for any two graphs G(1) and G(2) with eta(G(1)) = h and eta(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. 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 chi(G) + c', where c' is a constant. This improves the 2 log chi(G) + 3 bound in  2. Let G(1) and G(2) be two graphs such that chi(G1) >= chi(G2) >= clog(1.5)(chi(G(1))), where c is a constant. Then G1 square G2 satisfies Hadwiger's conjecture. 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 . ( They had shown that the Hadiwger's conjecture is true for G(d) if d >= 3).
|Item Type:||Journal Article|
|Additional Information:||Copyright of this article belongs to Springer.|
|Keywords:||Hadwiger Number;Hadwiger’s Conjecture;Graph Cartesian product;Minor;Chromatic number.|
|Department/Centre:||Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation)|
|Date Deposited:||23 Mar 2010 11:19|
|Last Modified:||19 Sep 2010 05:58|
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