Vertex-pancyclism in the generalized sum of digraphs

Abstract

A digraph D = (V, A) of order n ≥ 3 is pancyclic, whenever D contains a directed cycle of length k for each k ∈ {3, . . . , n}; and D is vertex-pancyclic iff, for each vertex v ∈ V and each k ∈ {3, . . . , n}, D contains a directed cycle of length k passing through v.Let D1, D2, . . . , D k be a collection of pairwise vertex disjoint digraphs. The generalized sum (g.s.) of D1, D2, . . . ,. . , k; and (iii) for each pair of vertices belonging to different summands of D, there is exactly one arc between them, with an arbitrary but fixed direction. A digraph D in ⊕ k i=1 Di will be called a generalized sum (g.s.) of D1, D2, . . . , D k .In this paper we prove that if D1 and D2 are two vertex disjoint Hamiltonian digraphs and D ∈ D1⊕D2 is strong, then at least one of the following assertions holds: D is vertex-pancyclic, it is pancyclic or it is Hamiltonian and contains a directed cycle of length l for each l ∈ {3, . . . , max{|V (Di)| + 1 : i ∈ {1, 2}}}. Moreover, we prove that if D1, D2, . . . , D k is a collection of pairwise vertex disjoint Hamiltonian digraphs, ni = |V (Di)| for each i ∈ {1, . . . , k} and D ∈ ⊕ k i=1 Di is strong, then at least one of the following assertions holds: D is vertex-pancyclic, it is pancyclic or it is Hamiltonian and contains a directed cycle of length l for each l ∈ {3, . . . , max{ i∈S ni + 1 : S ⊂ {1, . . . , k} with |S| = k − 1}}.