A note on the Path Kernel Conjecture

Let τ(G) denote the number of vertices in a longest path in a graph G = (V , E). A subset K of V is called a Pn-kernel of G if τ(G[K ]) ≤ n−1 and every vertex v ∈ V \K is adjacent to an end-vertex of a path of order n− 1 in G[K ]. It is known that every graph has a Pn-kernel for every positive integer n ≤ 9. R. Aldred and C. Thomassen in [R.E.L. Aldred, C. Thomassen, Graphs with not all possible path-kernels, Discrete Math. 285 (2004) 297–300] proved that there exists a graph which contains no P364-kernel. In this paper, we generalise this result. We construct a graph with no P155-kernel and for each integer l ≥ 0 we provide a construction of a graph G containing no Pτ(G)−l-kernel. © 2008 Elsevier B.V. All rights reserved.