Some results on the existence of Hamiltonian cycles in the generalized sum of digraphs

Abstract

Let D1,…,Dk be a family of pairwise vertex-disjoint digraphs. The generalized sum of D1,…,Dk, denoted by D1⊕⋯⊕Dk, is the set of all digraphs D which satisfies: (i) V(D)=∪i=1kV(Di), (ii) D[V(Di)]≅Di for every i∈[k] and (iii) between each pair of vertices in different summands of D there is exactly one arc. When each Di has no arcs, we have that D1⊕⋯⊕Dk is a set of k-partite digraphs. Moreover, for each tournament T on k vertices, we always have that T[D1,…,Dk], the composition of digraphs, is in D1⊕⋯⊕Dn. In this work, we give some results on the existence or the non-existence of Hamiltonian cycles and cycle-factors in such digraphs. Particularly, we prove a generalization of the characterization of Hamiltonian semicomplete bipartite digraphs due to Gutin, Häggkvist and Manoussakis.