On the Hadwiger's conjecture for graph products

Abstract

The Hadwiger number η ( G ) of a graph G is the largest integer h such that the complete graph on h nodes K h is a minor of G. Equivalently, η ( G ) is the largest integer such that any graph on at most η ( G ) nodes is a minor of G. The Hadwiger's conjecture states that for any graph G, η ( G ) ⩾ χ ( G ) , where χ ( G ) is the chromatic number of G. It is well-known that for any connected undirected graph G, there exists a unique prime factorization with respect to Cartesian graph products. If the unique prime factorization of G is given as G 1 □ G 2 □ ⋯ □ G k , where each G i is prime, then we say that the product dimension of G is k. Such a factorization can be computed efficiently. In this paper, we study the Hadwiger's conjecture for graphs in terms of their prime factorization. We show that the Hadwiger's conjecture is true for a graph G if the product dimension of G is at least 2 log 2 ( χ ( G ) ) + 3 . In fact, it is enough for G to have a connected graph M as a minor whose product dimension is at least 2 log 2 ( χ ( G ) ) + 3 , for G to satisfy the Hadwiger's conjecture. We show also that if a graph G is isomorphic to F d for some F, then η ( G ) ⩾ χ ( G ) ⌊ ( d - 1 ) / 2 ⌋ , and thus G satisfies the Hadwiger's conjecture when d ⩾ 3 . For sufficiently large d, our lower bound is exponentially higher than what is implied by the Hadwiger's conjecture. Our approach also yields (almost) sharp lower bounds for the Hadwiger number of well-known graph products like d-dimensional hypercubes, Hamming graphs and the d-dimensional grids. In particular, we show that for the d-dimensional hypercube H d , 2 ⌊ ( d - 1 ) / 2 ⌋ ⩽ η ( H d ) ⩽ 2 d / 2 d + 1 . We also derive similar bounds for G d for almost all G with n nodes and at least n log 2 n edges.