Eulerian and Hamiltonian dicycles in directed hypergraphs

Julio Araujo 1 Jean-Claude Bermond 2 Guillaume Ducoffe 2
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : In this article, we generalize the concepts of Eulerian and Hamiltonian digraphs to directed hypergraphs. A dihypergraph H is a pair (V(H), E(H)), where V(H) is a non-empty set of elements, called vertices, and E(H) is a collection of ordered pairs of subsets of V(H), called hyperarcs. It is Eulerian (resp. Hamiltonian) if there is a dicycle containing each hyperarc (resp. each vertex) exactly once. We first present some properties of Eulerian and Hamiltonian dihypergraphs. For example, we show that deciding whether a dihypergraph is Eulerian is an NP-complete problem. We also study when iterated line dihypergraphs are Eulerian and Hamiltonian. Finally, we study when the generalized de Bruijn dihypergraphs are Eulerian and Hamiltonian. In particular, we determine when they contain a complete Berge dicycle, i.e. an Eulerian and Hamiltonian dicycle.
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Julio Araujo, Jean-Claude Bermond, Guillaume Ducoffe. Eulerian and Hamiltonian dicycles in directed hypergraphs. Discrete Mathematics, Algorithms and Applications, World Scientific Publishing, 2014, 06, pp.1450012. ⟨10.1142/S1793830914500128⟩. ⟨hal-01104634v2⟩

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