ON PATH FACTORIZATIONS OF COMPLETE MULTIPARTITE GRAPHS

Necessary conditions for lambdaK(n, r), the complete multipartite grap h with r parts of size n in which each edge has multiplicity lambda, t o have a P(k)-factorization are nr = 0(mod k) and lambda(r-1)kn = 0(mo d 2(k-1)). We show that when n = 0(mod k) or r = 0(mod k), these two c onditions are also sufficient. (This implies that for all prime k the above two conditions are sufficient.) As corollaries, we also show tha t the necessary conditions are sufficient when r = 2 and r = 3.