Graphs with not all possible path-kernels

Abstract 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 ⩾ k ⩾2 there is a partition of V ( G ) into two parts (K, K ) 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 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.