Sparsest cuts and concurrent flows in product graphs

A cut [S.S] is a sparsest cut of a graph G if its cut value [S][S]/[S.S] is maximum (this is the reciprocal of the well-known edge-density of the cut). In the (undirected) uniform concurrent flow problem on G, between every vertex pair of G flow paths with a total flow of 1 have to be established. The objective is to minimize the maximum amount of flow through an edge (edge congestion). The minimum congestion value of the uniform concurrent flow problem on G is an upper bound for the maximum cut value of cuts in G. If both values are equal, G is called a bottleneck graph. The bottleneck properties of cartesian product graphs G×H are studied. First, a flow in G×H is constructed using optimal flows in G and H, and proven to be optimal. Secondly, two cuts are constructed in G×H using sparsest cuts of G and H. It is shown that one of these cuts is a sparsest cut of G×H. As a consequence, we can prove that G×H is (not) a bottleneck graph if both G and H are (not) bottleneck graphs.