Lexicographic Product of Extendable Graphs

Lexicographic product G o H of two graphs G and H has vertex set V(G) x V(H) and two vertices (u(1), v(1)) and (u(2), v(2)) are adjacent whenever u(1)u(2) is an element of E(G), or u(1) = u(2) and v(1)v(2) is an element of E(H). If every matching of G of size k can be extended to a perfect matching in G, then G is called k-extendable. In this paper, we study matching extendability in lexicographic product of graphs. The main result is that the lexicographic product of an m-extendable graph and an n-extendable graph is (m + 1)(n + 1)-extendable. In fact, we prove a slightly stronger result.