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arXiv · 1707.05252

Spanning Euler tours and spanning Euler families in hypergraphs with particular vertex cuts

Mateja Šajna·Yan D. Steimle
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Abstract

An Euler tour in a hypergraph is a closed walk that traverses each edge of the hypergraph exactly once, while an Euler family, first defined by Bahmanian and Sajna, is a family of closed walks that jointly traverse each edge exactly once and cannot be concatenated. In this paper, we study the notions of a spanning Euler tour and a spanning Euler family, that is, an Euler tour (family) that also traverses each vertex of the hypergraph at least once. We examine necessary and sufficient conditions for a hypergraph to admit a spanning Euler family, most notably, when the hypergraph possesses a vertex cut consisting of vertices of degree two. Moreover, we characterise hypergraphs with a vertex cut of cardinality at most two that admit a spanning Euler tour (family). This result enables us to reduce the problem of existence of a spanning Euler tour (which is NP-complete), as well as the problem of a spanning Euler family, to smaller hypergraphs.

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Mateja Šajna, Yan D. Steimle. 2017-07-17. Spanning Euler tours and spanning Euler families in hypergraphs with particular vertex cuts. https://arxiv.org/abs/1707.05252

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