Abstract
The detour order of a graph G , denoted by τ ( G ) , is the order of a longest path in G . The Path Partition Conjecture (PPC) is the following: If G is any graph and ( a , b ) any pair of positive integers such that τ ( G ) = a + b , then the vertex set of G has a partition ( A , B ) such that τ ( 〈 A 〉 ) ⩽ a and τ ( 〈 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.
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