An Asymptotic Result for the Path Partition Conjecture

The detour order of a graph G, denoted by τ (G) , is the order of a longest path in G. A partition of the vertex set of G into two sets, A and B, such that τ(〈A〉) ≤ a and τ(〈B〉) ≤ b is called an (a, b)-partition of G. If G has an (a, b)-partition for every pair (a, b) of positive integers such that a + b = τ(G), then we say that G is τ -partitionable. The Path Partition Conjecture (PPC), which was discussed by Lovász and Mihók in 1981 in Szeged, is that every graph is τ -partitionable. It is known that a graph G of order n and detour order τ = n − p is τ -partitionable if p = 0, 1. We show that this is also true for p = 2, 3, and for all p ≥ 4 provided that n ≥ p(10p − 3).