The Path Partition Conjecture is true for claw-free graphs

The detour order of a graph GG, denoted by τ(G)τ(G), is the order of a longest path in GG. The Path Partition Conjecture (PPC) is the following: If GG is any graph and (a,b)(a,b) any pair of positive integers such that τ(G)=a+bτ(G)=a+b, then the vertex set of GG has a partition (A,B)(A,B) such that τ(〈A〉)⩽aτ(〈A〉)⩽a and τ(〈B〉)⩽bτ(〈B〉)⩽b. We prove that this conjecture is true for the class of claw-free graphs. We also show that to prove that the PPC is true, it is sufficient to consider the class of 2-connected graphs.