Hadwiger Number and the Cartesian Product Operation on Graphs

The Hadwiger number η(G) of a graph G is defined as the largest integer n for which the complete graph on n nodes Kn is a minor of G. Hadwiger conjectured that for any graph G, η(G) ≥ χ(G),where χ(G) is the chromatic number of G. In this paper, we investigate the Hadwiger number with respect to the cartesian product operation on Graphs. As the main result of this paper, we show that for any two graphs G1 and G2 with η(G1) = h and η(G2) = l, η(G1 2G2) ≥ 1 4 (h − √ l)( √ l − 2). (Since G1 2G2 is isomorphic to G2 2G1, we can assume without loss of generality that h ≥ l). This lower bound is the best possible (up to a small constant factor), since if G1 = Kh and G2 = Kl, η(G1 2G2) ≤ 2h √ l. We also show that η(G1 2G2) doesn’t have any upper bound which depends only on η(G1) and η(G2), by demonstrating graphs G1 and G2 such that η(G1) and η(G2) are bounded whereas η(G1 2G2) grows with the number of nodes. (The problem of studying the Hadwiger number with respect to the cartesian product operation was posed by Z.Miller in 1978.) As consequences of our main result, we show the following: 1. Let G be a connected graph. Let the (unique) prime factorization of G be given by G1 2G2 2 ...2Gk . 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 given in [2]. 2. Let G1 and G2 be two graphs such that χ(G1) ≥ χ(G2) ≥ clog(χ(G1)), where c is a constant. Then G1 2G2 satisfies Hadwiger’s conjecture. In fact, in the special case where χ(G1) = χ(G2) we give a different (and simpler) proof to show that G1 2G2 satisfies Hadwiger’s conjecture. As a consequence we show that Hadwiger’s conjecture is true for G (cartesian product of G taken d times) for any graph G, if d ≥ 2. This settles a question asked by Chandran and Sivadasan [2]. (They had shown that Hadiwger’s conjecture is true for G for any graph G, if d ≥ 3.) We also improve a lower bound proved by Chandran and Sivadasan on the Hadwiger number of Hamming graphs.