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On the connectivity of Cartesian product of graphs

We give a new alternative proof of Liouville’s formula which states that for any graphs G and H on at least two vertices, κ ( G □  H ) = min{ κ ( G )| H |,  | G | κ ( H ),   δ ( G ) +  δ ( H )} , where κ and δ denote the connectivity number and minimum degree of a given graph, respectively. The main idea of our proof is based on construction of a vertex-fan which connects a vertex from V ( G □  H ) to a subgraph of G □  H . We also discuss the edge version of this problem as well as formula for products with more than two factors.