The Menger number of the strong product of graphs

The xy-Menger number with respect to a given integer ℓ, for every two vertices x,y in a connected graph G, denoted by ζℓ(x,y), is the maximum number of internally disjoint xy-paths whose lengths are at most ℓ in G. The Menger number of G with respect to ℓ is defined as ζℓ(G)=min{ζℓ(x,y):x,y∈V(G)}. In this paper we focus on the Menger number of the strong product G1⊠G2 of two connected graphs G1 and G2 with at least three vertices. We show that ζℓ(G1⊠G2)≥ζℓ(G1)ζℓ(G2) and furthermore, that ζℓ+2(G1⊠G2)≥ζℓ(G1)ζℓ(G2)+ζℓ(G1)+ζℓ(G2) if both G1 and G2 have girth at least 5. These bounds are best possible, and in particular, we prove that the last inequality is reached when G1 and G2 are maximally connected graphs.