Triple Systems are Eulerian

Abstract

An Euler tour of 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, using a graph-theoretic approach, we prove that every triple system with at least two triples is eulerian, that is, it admits an Euler tour. Horan and Hurlbert have previously shown that for every admissible order >3, there exists a Steiner triple system with an Euler tour, while Dewar and Stevens have proved that every cyclic Steiner triple system of order >3 and every cyclic twofold triple system admits an Euler tour.