On the Strong Product of a k-Extendable and an l-Extendable Graph

Abstract.

 Let G ₁⊗G ₂ be the strong product of a k-extendable graph G ₁ and an l-extendable graph G ₂, and X an arbitrary set of vertices of G ₁⊗G ₂ with cardinality 2[(k+1)(l+1)/2]. We show that G ₁⊗G ₂−X contains a perfect matching. It implies that G ₁⊗G ₂ is [(k+1)(l+1)/2]-extendable, whereas the Cartesian product of G ₁ and G ₂ is only (k+l+1)-extendable. As in the case of the Cartesian product, the proof is based on a lemma on the product of a k-extendable graph G and K ₂. We prove that G⊗K ₂ is (k+1)-extendable, and, a bit surprisingly, it even remains (k+1)-extendable if we add edges to it.