Vertex pancyclicity over lexicographic products

A lexicographic product of graphs G and H, denoted by G omicron H, is defined as a graph with the vertex set V(G) x V(H) and an edge {(u(1), v(1)), (u(2), v(2))} presents in the product 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)). We investigate the sufficient conditions for vertex pancyclicity of lex- icographic products of complete graphs K-n, paths P-n or cycles C-n with a general graph. We obtain that (i) if G(1) is a traceable graph of even order and G(2) is a graph with at least one edge, then G(1) omicron G(2) is vertex pancyclic; (ii) if G(1) is hamiltonian and G(2) is a graph with at least one edge, then G(1) omicron G(2) is vertex pancyclic.