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 $cell^4sqrt{logell}$.

Blocking Coloured Point Sets

This paper studies problems related to visibility among points in the plane. A point $x$ emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $bar{vw}$. A set of points $P$ is emph{$k$-blocked} if each point in $P$ is assigned one of $k$ colours, such that distinct points $v,win P$ are assigned the same colour if and only if some other point in $P$ blocks $v$ and $w$. The focus of this paper is the conjecture that each $k$-blocked set has bounded size (as a function of $k$). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets ${n_1,n_2,n_3,n_4}$ such that some 4-blocked set has exactly $n_i$ points in the $i$-th colour class. Amongst other results, for infinitely many values of $k$, we construct $k$-blocked sets with $k^{1.79...}$ points.

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem: for every integer $ell geq 2$, every sufficiently large set of points in the plane contains $ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the big line or big clique conjecture of Kara, Por, and Wood [emph{Discrete Comput. Geom.} 34(3):497--506, 2005].

Distinct Distances in Graph Drawings

The emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in $mathcal{O}(log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distance-number. Moreover, as $Delta$ increases the existential lower bound on the distance-number of $Delta$-regular graphs tends to $Omega(n^{0.864138})$.