Bounded edge-connectivity and edge-persistence of Cartesian product of graphs

The bounded edge-connectivity λk(G)λk(G) of a connected graph GG with respect to k(≥d(G)) is the minimum number of edges in GG whose deletion from GG results in a subgraph with diameter larger than kk and the edge-persistence D+(G)D+(G) is defined as λd(G)(G)λd(G)(G), where d(G)d(G) is the diameter of GG. This paper considers the Cartesian product G1×G2G1×G2, shows λk1+k2(G1×G2)≥λk1(G1)+λk2(G2)λk1+k2(G1×G2)≥λk1(G1)+λk2(G2) for k1≥2k1≥2 and k2≥2k2≥2, and determines the exact values of D+(G)D+(G) for G=Cn×PmG=Cn×Pm, Cn×CmCn×Cm, Qn×PmQn×Pm and Qn×CmQn×Cm.