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 h(G) is 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 Hadwigers Conjecture, which states that the chromatic number chi(G) <= h(G). It is open whether Hadwigers Conjecture holds for every Cartesian product. We prove that if |V(H)|-1 >= chi(G) >= chi(H) then Hadwigers Conjecture holds for G*H. On the other hand, we prove that Hadwigers Conjecture holds for all Cartesian products if and only if it holds for all G * K_2. We then show that h(G * K_2) 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).

Polynomial treewidth forces a large grid-like-minor

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $elltimesell$ grid minor is exponential in $ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A emph{grid-like-minor of order} $ell$ in a graph $G$ is a set of paths in $G$ whose intersection graph is bipartite and contains a $K_{ell}$-minor. For example, the rows and columns of the $elltimesell$ grid are a grid-like-minor of order $ell+1$. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least $cell^4sqrt{logell}$ has a grid-like-minor of order $ell$. As an application of this result, we prove that the cartesian product $Gsquare K_2$ contains a $K_{ell}$-minor whenever $G$ has treewidth at least $cell^4sqrt{logell}$.