On the Hadwiger's conjecture for graph products

The Hadwiger number η(G)η(G) of a graph G is the largest integer h such that the complete graph on h   nodes KhKh is a minor of G  . Equivalently, η(G)η(G) is the largest integer such that any graph on at most η(G)η(G) nodes is a minor of G. The Hadwiger's conjecture states that for any graph G  , η(G)⩾χ(G)η(G)⩾χ(G), where χ(G)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 G1□G2□⋯□GkG1□G2□⋯□Gk, where each GiGi 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 2log2(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 2log2(χ(G))+3, for G to satisfy the Hadwiger's conjecture. We show also that if a graph G   is isomorphic to FdFd for some F  , then η(G)⩾χ(G)⌊(d-1)/2⌋η(G)⩾χ(G)⌊(d-1)/2⌋, and thus G   satisfies the Hadwiger's conjecture when d⩾3d⩾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 HdHd, 2⌊(d-1)/2⌋⩽η(Hd)⩽2d/2d+1. We also derive similar bounds for GdGd for almost all G with n   nodes and at least nlog2n edges.