Covering hypergraphs are eulerian

TL;DR

This paper proves that all covering k-hypergraphs with at least two edges have an Euler tour, extending the understanding of Eulerian properties in hypergraph structures.

Contribution

It establishes a necessary and sufficient condition for the existence of Euler tours in covering hypergraphs for all k ≥ 3.

Findings

Every covering k-hypergraph with ≥ 2 edges admits an Euler tour.

The condition is both necessary and sufficient for k ≥ 3.

This extends Eulerian theory to a broad class of hypergraphs.

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 $k$-hypergraph to be a non-empty $k$-uniform hypergraph in which every $(k-1)$-subset of vertices appear together in at least one edge. We then show that every covering $k$-hypergraph, for $k\geq 3$, admits an Euler tour if and only if it has at least two edges.