Abstract
Abstract The detour order τ(G) of a graph G is the order of a longest path of G. A partition (A, B) of V is called an (a, b)-partition of G if τ(G[A]) ≤ a and τ(G[B]) ≤ b. The Path Partition Conjecture is the following: For any graph G, with detour order τ(G) = a + b, there exists an (a, b)-partition of G. We introduce and examine a conjecture which is possibly stronger: If M is a maximum Pn+1-free set of vertices of G, with n
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