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 characteri...

In this note connected, edge-transitive lexicographic and Cartesian products are characterized. For the lexicographic product G ◦H of a connected graph G that is not complete by a graph H, we show that it is edge-transitive if and only if G is edge-transitive and H is edgeless. If the first factor of G ◦H is non-trivial and complete, then G ◦H is edge-transitive if and only if H is the lexicogr...

In this article we determine the crossing numbers of the Cartesian products of given three graphs on five vertices with paths.

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On ...

Let H be a graph on n vertices and G a collection of n subgraphs of H, one for each vertex, where G is an orthogonal double cover (ODC) of H if every edge of H occurs in exactly two members of G and any two members share an edge whenever the corresponding vertices are adjacent in H and share no edges whenever the corresponding vertices are nonadjacent in H. In this paper, we are concerned with ...

We give a characterization for isoperimetric invariants, including the Cheeger constant and the isoperimetric number of a graph. This leads to an isoperimetric inequality for the cartesian products of graphs.

The edge-integrity of a graph measures the difficulty of breaking it into pieces through the removal of a set of edges, taking into account both the number of edges removed and the size of the largest surviving component. We develop some techniques for bounding, estimating, and computing the edgeintegrity of products of graphs, paying particular attention to grid graphs. Correspondence to: Wayn...

The Cartesian product of graphs was introduced more than 50 years ago and many fundamental results were obtained since then. Nevertheless, in the last years several basic problems on the Cartesian product were solved and interesting theorems proved on topics of contemporary interest in graph theory. Here we survey recent developments on the structure of the Cartesian product with emphasis on th...

We prove that the Seidel morphism of (M × M , ω ⊕ ω) is naturally related to the Seidel morphisms of (M,ω) and (M , ω), when these manifolds are monotone. We deduce a condition for loops of Hamiltonian diffeomorphisms of the product to be homotopically non trivial. This result was inspired by and extends results obtained by Pedroza [P]. All the symplectic manifolds we consider in this note are ...

The projective tensor product in a category of topologicalR-modules (where R is a topological ring) can be defined in Top, the category of topological spaces, by the same universal property used to define the tensor product of R-modules in Set. In this article, we extend this definition to an arbitrary topological category X and study how the cartesian closedness of X is related to the monoidal...