@MISC{Wood08cliqueminors, author = {David R. Wood}, title = {CLIQUE MINORS IN CARTESIAN PRODUCTS OF GRAPHS}, year = {2008} }

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Abstract

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 ηÔGÕis the maximum cardinality of a clique minor in G. 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 states that the chromatic number χÔGÕ�ηÔGÕ. It is open whether Hadwiger’s Conjecture holds for every Cartesian product. We prove that if�VÔHÕ�¡1�χÔGÕ�χÔHÕthen Hadwiger’s Conjecture holds for G¥H. 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