Graphs with not all possible path-kernels

The Path Partition Conjecture states that the vertices of a graph G with longest path of length c may be partitioned into two parts X and Y such that the longest path in the subgraph of G induced by X has length at most a and the longest path in the subgraph of G induced by Y has length at most b, where a + b = c. Moreover, for each pair a, b such that a + b = c there is a partition with this property. A stronger conjecture by Broere, Hajnal and Mihok, raised as a problem by Mihok in 1985, states the following: For every graph G and each integer k, c greater than or equal to k greater than or equal to 2 there is a partition of V(G) into two parts (K, (K) over bar) such that the subgraph G[K] of G induced by K has no path on more than k - 1 vertices and each vertex in (K) over bar is adjacent to an endvertex of a path on k - 1 vertices in G[K]. In this paper we provide a counterexample to this conjecture. (C) 2004 Elsevier B.V. All rights reserved.