On Tours that Contain All Edges of a Hypergraph

Let H be a k-uniform hypergraph, k > 2. By an Euler tour in H we mean an alternating sequence v0, e1, v1, e2, v2, . . . , vm−1, em, vm = v0 of vertices and edges in H such that each edge of H appears in this sequence exactly once and vi−1, vi ∈ ei, vi−1 6= vi, for every i = 1, 2, . . . ,m. This is an obvious generalization of the graph theoretic concept of an Euler tour. A straightforward necessary condition for existence of an Euler tour in a k-uniform hypergraph is |Vodd(H)| 6 (k − 2)|E(H)|, where Vodd(H) is the set of vertices of odd degrees in H and E(H) is the set of edges in H. In this paper we show that this condition is also sufficient for hypergraphs of a broad class of k-uniform hypergraphs, that we call strongly connected hypergraphs. This result reduces to the Euler theorem on existence of Euler tours, when k = 2, i.e. for graphs, and is quite simple to prove for k > 3. Therefore, we concentrate on the most interesting case of k = 3. In this case we further consider the problem of existence of an Euler tour in a certain class of 3-uniform hypergraphs containing the class of strongly connected hypergraphs as a proper subclass. For hypergraphs in this class we give a sufficient condition for existence of an Euler tour and prove intractability (NP-completeness) of the problem in this class in general.