Arbitrary Partitionability Of Three Kinds Of Product Graphs

An n-vertex graph G is called arbitrarily partitionable(AP,for short),if for any sequence ?=(?1,?2,…,?k)of positive integers such that(?),there exists a partition(V1,V2,…,Vk)of the vertex set V(G)such that for all 1?i?k,|Vi|=?i and the subgraph G[Vi]induced by Vi is connected.In this paper,we mainly discuss the arbitrary partitionability of product graphs.For the Cartesian product,we mainly discuss the arbitrary partitionability of K1,m?Pn,S*?Pm and S**?Pm,where S*and S**are two types of generalized sun-like graphs.For the Direct product of H×Cn,we study the arbitrarily partitionability for H ?{Pm,Cm,K1,m?.For the lexicographic product,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 o Pn is AP;if G is a traceable graph and H is an AP graph,then G o H is AP;if G=S(2,a,b)is an AP star-like tree with 2?a?b,then G o G is AP;if G is Hamiltonian,H is a graph,then G o H is AP.