Partitions of a graph into paths with prescribed endvertices and lengths

For a graph G, let sigma(2)(G) denote the minimum degree sum of a pair of n onadjacent vertices. We conjecture that if \V(G)\ = n = Sigma(i)(k) = 1 a(i ) and sigma(2)(G) greater than or equal to n + k- 1, then for any k vertice s v(1), v(2),...,v(k) in G, there exist vertex-disjoint paths P-1,P-2,...,P -k such that \V(P-i)\ = a(i) and v(i) is an endvertex of P-i for 1 less tha n or equal to i less than or equal to k. In this paper, we verify the conje cture for the cases where almost all a(i) less than or equal to 5, and the cases where k less than or equal to 3. (C) 2000 John Wiley & Sons, Inc.