An n-vertex graph G is said to be arbitrarily partitionable(AP

for short)

if for any sequence (n1

n2

···

nk) of positive integers such that ■ ni= n

there exists a partition(V1

V2

···

Vk) of vertex set V(G)such that the subgraph G[Vi] induced by Viis connected and |Vi| = nifor each 1 ≤ i ≤ k. The lexicographic product of two graphs G and H

denoted by G ? H

has vertex set V(G) × V(H) in which(g

h)(g

h) is an edge whenever gg is an edge in G or g = g and hh is an edge in H. In this paper

we mainly discuss the arbitrary partitionability of the lexicographic product of traceable graph and AP graph. We prove that for a tree T of maximum degree at most n+1

if T has a path P such that all the vertices of degree(T) are in V(P)

then the lexicographic product graph T ? Pnis AP; if G is a traceable graph and H is an AP graph

then G ? H is AP; if G = S(2

a

b) is an AP star-like tree with 2 ≤ a ≤ b

then G ? G is AP; if G is Hamiltonian

H is a graph

then G ? H is AP.