Covering hypergraphs are Eulerian. Issue 2 (15th March 2022) - Šajna, Mateja; Wagner, Andrew
Abstract: An Euler tour in a hypergraph (also called a rank‐2 universal cycle or 1‐overlap cycle in the context of designs) is a closed walk that traverses every edge exactly once. In this paper, we define a covering k k ‐ hypergraph to be a nonempty k k ‐uniform hypergraph in which every ( k − 1 ) (k-1) ‐subset of vertices appear together in at least one edge. We then show that every covering k k ‐hypergraph, for k ≥ 3 k3, admits an Euler tour if and only if it has at least two edges.