Hadwiger Number and the Cartesian Product of Graphs

The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph Kn 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 H of graphs. As the main result of this paper, we prove that η(G1 G2) ≥ h √ l (1− o(1)) for any two graphs G1 and G2 with η(G1) = h and η(G2) = 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 = G1 G2 ... Gk 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. Let G1 and G2 be two graphs such that χ(G1) ≥ χ(G2) ≥ c log(χ(G1)), where c is a constant. Then G1 G2 satisfies Hadwiger’s conjecture. 3. Hadwiger’s conjecture is true for Gd (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 Gd if d ≥ 3).