On the edge-connectivity and restricted edge-connectivity of a product of graphs

The product graph Gm*GpGm*Gp of two given graphs GmGm and GpGp was defined by Bermond et al. [Large graphs with given degree and diameter II, J. Combin. Theory Ser. B 36 (1984) 32–48]. For this kind of graphs we provide bounds for two connectivity parameters (λλ and λ′λ′, edge-connectivity and restricted edge-connectivity, respectively), and state sufficient conditions to guarantee optimal values of these parameters. Moreover, we compare our results with other previous related ones for permutation graphs and cartesian product graphs, obtaining several extensions and improvements. In this regard, for any two connected graphs GmGm, GpGp of minimum degrees δ(Gm)δ(Gm), δ(Gp)δ(Gp), respectively, we show that λ(Gm*Gp)λ(Gm*Gp) is lower bounded by both δ(Gm)+λ(Gp)δ(Gm)+λ(Gp) and δ(Gp)+λ(Gm)δ(Gp)+λ(Gm), an improvement of what is known for the edge-connectivity of Gm×GpGm×Gp.