Constructing edge-disjoint Steiner trees in Cartesian product networks

Cartesian product networks are always regarded as a tool for ``combining'' two given networks with established properties to obtain a new one that inherits properties from both. For a graph $F=(V,E)$ and a set S⊆ V(F) of at least two vertices, an S-Steiner tree or a Steiner tree connecting S (or simply, an S-tree) is a subgraph $T=(V',E')$ of F that is a tree with S⊆ V'. For S⊆ V(F) and |S|≥ 2, the { generalized local edge-connectivity} $λ(S)$ is the maximum number of edge-disjoint Steiner trees connecting S in F. For an integer k with 2≤ k≤ n, the { generalized k-edge-connectivity} λₖ(F) of a graph F is defined as λₖ(F)=min\λ(S)\,|\,S⊆ V(F) \ and \ |S|=k\.In this paper, we give sharp upper and lower bounds for λₖ(G H), where is the Cartesian product operation, and $G,H$ are two graphs. 14 pages; 3 figures