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

Let G1 ⊗ G2 be the strong product of a k-extendable graph G1 and an l-extendable graph G2, and X an arbitrary set of vertices of G1⊗G2 with cardinality 2[(k + 1)(l + 1)/2]. We show that G1 ⊗ G2 − X contains a perfect matching. It implies that G1 ⊗ G2 is [(k + 1)(l + 1)/2]-extendable, whereas the Cartesian product of G1 and G2 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 K2. We prove that G⊗K2 is (k + 1)-extendable, and, a bit surprisingly, it even remains (k + 1)-extendable if we add edges to it.