The Menger number of the strong product of graphs

The x y -Menger number with respect to a given integer ℓ , for every two vertices x , y in a connected graph G , denoted by ζ ℓ ( x , y ) , is the maximum number of internally disjoint x y -paths whose lengths are at most ℓ in G . The Menger number of G with respect to ℓ is defined as ζ ℓ ( G ) = min { ζ ℓ ( x , y ) : x , y ∈ V ( G ) } . In this paper we focus on the Menger number of the strong product G 1 ⊠ G 2 of two connected graphs G 1 and G 2 with at least three vertices. We show that ζ ℓ ( G 1 ⊠ G 2 ) ≥ ζ ℓ ( G 1 ) ζ ℓ ( G 2 ) and furthermore, that ζ ℓ + 2 ( G 1 ⊠ G 2 ) ≥ ζ ℓ ( G 1 ) ζ ℓ ( G 2 ) + ζ ℓ ( G 1 ) + ζ ℓ ( G 2 ) if both G 1 and G 2 have girth at least 5. These bounds are best possible, and in particular, we prove that the last inequality is reached when G 1 and G 2 are maximally connected graphs.