2023/01/26 by Li, Rui, Gutin, Gregory, Zhang, He +3
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2301.12933
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 \it 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 \it 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\Box H), where \Box is the Cartesian product operation, and G,H are two graphs.