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

TL;DR

This paper investigates conditions under which hypergraphs admit spanning Euler tours and families, especially focusing on vertex cuts of small size, and reduces the complexity of these problems.

Contribution

It provides necessary and sufficient conditions for spanning Euler tours and families in hypergraphs with small vertex cuts, simplifying their existence verification.

Findings

Characterizes hypergraphs with small vertex cuts admitting spanning Euler structures

Reduces the problem of existence to smaller hypergraphs

Provides conditions for hypergraphs with vertex cuts of size at most two

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.