Path partitions and [formula omitted]-free sets

Abstract

The detour order τ ( G ) of a graph G is the order of a longest path of G. If S is a subset of V ( G ) such that the graph induced by S has detour order at most n, then S is called a P n + 1 - free set in G. The Path Partition Conjecture (PPC) can be stated as follows: For any graph G and any positive integer n < τ ( G ) , there exists a P n + 1 -free set H in G such that τ ( G - H ) ⩽ τ ( G ) - n . We prove that if G is any graph and M is any maximal P n + 1 -free set in G, then τ ( G - M ) ⩽ τ ( G ) - 2 3 ( n + 1 ) . We also prove that if G has no cycle of order less than n or greater than τ ( G ) - n + 2 , then τ ( G - M ) ⩽ τ ( G ) - n for every maximal P n + 1 -free subset M of G . As a corollary of the latter result we prove that the PPC is true for the class of connected, weakly pancyclic graphs.