 

David R. Wood
Clique minors in Cartesian products of graphs view print


Published: 
September 30, 2011 
Keywords: 
Graph minor, Cartesian product, Hadwiger number 
Subject: 
graph minors 05C83, structural characterization of types of graphs 05C75 


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 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 ϗ(G)≦η(G). It is open whether Hadwiger's Conjecture holds for every Cartesian product. We prove that G □ H (where ϗ(G)≧ϗ(H)) satisfies Hadwiger's Conjecture whenever:
 H has at least ϗ(G)+1 vertices, or
 the treewidth of G is sufficiently large compared to ϗ(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 □ K_{2}. We then show that η(G □ K_{2}) is tied to the treewidth of G.
We also develop connections with pseudoachromatic colourings and
connected dominating sets that imply neartight bounds on the Hadwiger
number of grid graphs (Cartesian products of paths) and Hamming graphs
(Cartesian products of cliques).


Acknowledgements
Supported by QEII Research Fellowship from the Australian Research Council. Research initiated at the Universitat Polit{è}cnica de Catalunya (Barcelona, Spain) where supported by a Marie Curie Fellowship of the European Commission under contract MEIFCT2006023865, and by the projects MEC MTM200601267 and DURSI 2005SGR00692.


Author information
Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia
woodd@unimelb.edu.au

