Generalized 3-edge-connectivity of Cartesian product graphs

The generalized k-connectivity κk(G) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k-edge-connectivity which is defined as λk(G) = min{λ(S): S ⊆ V (G) and |S| = k}, where λ(S) denotes the maximum number l of pairwise edge-disjoint trees T1, T2, …, Tl in G such that S ⊆ V (Ti) for 1 ⩽ i ⩽ l. In this paper we prove that for any two connected graphs G and H we have λ3(G □ H) ⩾ λ3(G) + λ3(H), where G □ H is the Cartesian product of G and H. Moreover, the bound is sharp. We also obtain the precise values for the generalized 3-edge-connectivity of the Cartesian product of some special graph classes.