Clique Minors in Cartesian Products of Graphs

A clique minor in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The Hadwiger number η(G) is the maximum cardinality of a clique minor in G. It is one of the principle measures of the structural complexity of a graph. This paper studies clique minors in the Cartesian product G H. Our main result is a rough structural characterisation theorem for Cartesian products with bounded Hadwiger number. It implies that if the product of two sufficiently large graphs has bounded Hadwiger number then it is one of the following graphs: • a planar grid with a vortex of bounded width in the outerface, • a cylindrical grid with a vortex of bounded width in each of the two ‘big’ faces, or • a toroidal grid. Motivation for studying the Hadwiger number of a graph includes Hadwiger’s Conjecture, which asserts that the chromatic number χ(G) ≤ η(G). It is open whether Hadwiger’s Conjecture holds for every Cartesian product. We prove that G H (where χ(G) ≥ χ(H)) satisfies Hadwiger’s Conjecture whenever: • H has at least χ(G) + 1 vertices, or • the treewidth of G is sufficiently large compared to χ(G). On the other hand, we prove that Hadwiger’s Conjecture holds for all Cartesian products if and only if it holds for all G K2. We then show that η(G K2) is tied to the treewidth of G. We also develop connections with pseudoachromatic colourings and connected dominating sets that imply near-tight bounds on the Hadwiger number of grid graphs (Cartesian products of paths) and Hamming graphs (Cartesian products of cliques). Received March 27, 2008; accepted: September 3, 2011; revised: September 22, 2011. 2010 Mathematics Subject Classification. graph minors 05C83, structural characterization of types of graphs 05C75.