A note on the Path Kernel Conjecture

Let τ(G)τ(G) denote the number of vertices in a longest path in a graph G=(V,E)G=(V,E). A subset KK of VV is called a PnPn-kernel   of GG if τ(G[K])≤n−1τ(G[K])≤n−1 and every vertex v∈V∖Kv∈V∖K is adjacent to an end-vertex of a path of order n−1n−1 in G[K]G[K]. It is known that every graph has a PnPn-kernel   for every positive integer n≤9n≤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 P364P364-kernel. In this paper, we generalise this result. We construct a graph with no P155P155-kernel and for each integer l≥0l≥0 we provide a construction of a graph GG containing no Pτ(G)−lPτ(G)−l-kernel.