The Path Partition Conjecture for Oriented Graphs

The vertex set and arc set of a digraph D are denoted by V (D) and E (D), respectively, and the number of vertices in a digraph D is denoted by n (D). A directed cycle (path, walk) in a digraph will simply be called a cycle (path, walk). A graph or digraph is called hamiltonian if it contains a cycle that visits every vertex, traceable if it contains a path that visits every vertex, and walkable if it contains a walk that visits every vertex. A digraph D is called strong (or strongly connected) if every vertex of D is reachable from every other vertex. Thus a digraph D of order bigger than 1 is strong if and only if it contains a closed walk that visits every vertex. A maximal strong subdigraph of a digraph D is called a strong component of D and a maximal walkable subdigraph of D is called a walkable component of D. A longest path in a digraph D is called a detour of D. The order of a detour of D is called the detour order of D and is denoted by λ (D) . If (a, b) is a pair of positive integers, a partition (A,B) of the vertex set of a digraph D is called an (a, b)-partition if λ(D〈A〉) ≤ a and λ(D〈B〉) ≤ b. If a digraph D has an (a, b)-partition for every pair of positive integers (a, b) such that a + b = λ (D), then D is called λ-partitionable. The Directed Path Partition Conjecture (DPPC) states: