EULERIAN AND HAMILTONIAN DICYCLES IN DIRECTED HYPERGRAPHS

Author:

ARAUJO JULIO1,BERMOND JEAN-CLAUDE1,DUCOFFE GUILLAUME2

Affiliation:

1. COATI Project, INRIA and I3S (CNRS/UNS), 2004, Route des Lucioles, B.P. 93, F-06902 Sophia Antipolis Cedex, France

2. University of Nice-Sophia Antipolis, France

Abstract

In this paper, we generalize the concepts of Eulerian and Hamiltonian digraphs to directed hypergraphs. A dihypergraphH 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.

Publisher

World Scientific Pub Co Pte Lt

Subject

Discrete Mathematics and Combinatorics

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