Graph partition into paths containing specified vertices

For a graph G, let 2(G) denote the minimum degree sum of a pair of nonadjacent vertices. Suppose G is a graph of order n. Enomoto and Ota (J. Graph Theory 34 (2000) 163–169) conjectured that, if a partition n = ∑k i=1 ai is given and 2(G)¿ n + k − 1, then for any k distinct vertices v1; : : : ; vk , G can be decomposed into vertex-disjoint paths P1; : : : ; Pk such that |V (Pi)| = ai and vi is an endvertex of Pi. Enomoto and Ota (J. Graph Theory 34 (2000) 163) veri!ed the conjecture for the case where all ai 6 5, and the case where k6 3. In this paper, we prove the following theorem, with a stronger assumption of the conjecture. Suppose G is a graph of order n. If a partition n= ∑k i=1 ai is given and 2(G)¿ ∑k i=1 max( 3ai ; ai +1)−1, then for any k distinct vertices v1; : : : ; vk , G can be decomposed into vertex-disjoint paths P1; : : : ; Pk such that |V (Pi)|=ai and vi is an endvertex of Pi for all i. This theorem implies that the conjecture is true for the case where all ai 6 5 which was proved in (J. Graph Theory 34 (2000) 163–169). c © 2002 Elsevier Science B.V. All rights reserved.