Quasi-Eulerian hypergraphs. (English) Zbl 1369.05153

Bahmanian, Amin; Šajna, Mateja

Electron. J. Comb. 24, No. 3, Research Paper P3.30, 12 p. (2017).

Summary: We generalize the notion of an Euler tour in a graph in the following way. An Euler family in a hypergraph is a family of closed walks that jointly traverse each edge of the hypergraph exactly once. An Euler tour thus corresponds to an Euler family with a single component. We provide necessary and sufficient conditions for the existence of an Euler family in an arbitrary hypergraph, and in particular, we show that every 3-uniform hypergraph without cut edges admits an Euler family. Finally, we show that the problem of existence of an Euler family is polynomial on the class of all hypergraphs.
This work complements existing results on rank-1 universal cycles and 1-overlap cycles in triple systems, as well as recent results by Z. Lonc and P. Naroski [Electron. J. Comb. 17, No. 1, Research Paper R144, 31 p. (2010; Zbl 1204.05064)], who showed that the problem of existence of an Euler tour in a hypergraph is NP-complete.

MSC:

05C65	Hypergraphs
05C45	Eulerian and Hamiltonian graphs
05C70	Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05B07	Triple systems

Keywords:

hypergraph; Euler tour; Eulerian hypergraph; Euler family; quasi-Eulerian hypergraph; \((g,f)\)-factor