Euler tours in hypergraphs (Q2658380)

Let \(G\) be a \(k\)-uniform hypergraph. A sequence of vertices \({\mathcal{W}} = x_1x_2\dots x_{\ell}\) is a (tight self-avoiding) walk in \(G\) if \(\{x_i,x_{i+1}, \dots, x_{i+k-1} \} \in E(G)\) for all \(i \in \{1,2,\dots, \ell-k+1\}\). \(\mathcal{W}\) is a closed walk if \(\{x_i,x_{i+1}, \dots,x_{i+k-1}\} \in E(G)\) for all \(i \in \{1,2,\dots, \ell\}\), with indices modulo \(\ell\), and no edge of \(G\) appears more than once in this way. Suppose \(E(W)\) denotes the set of edges appearing in a walk \(W\). An Euler tour of \(G\) is a closed walk \(W\) in \(G\) with \(E(W)=E(G)\). In this paper, the authors show that a quasirandom \(k\)-uniform hypergraph \(G\) has a tight Euler tour subject to the necessary condition that \(k\) divides all vertex degrees. As a consequence they also prove an old conjecture due to \textit{F. Chung} et al. [Discrete Math. 110, No. 1--3, 43--59 (1992; Zbl 0776.05001)] on the existence of universal cycles for the \(k\)-subsets of an \(n\)-set. The proof of the main result uses recent work on \(F\)-decompositions of \(k\)-uniform hypergraphs due to \textit{S. Glock} at al. [``The existence of designs via iterative absorption: hypergraph \(F\)-designs for arbitrary \(F\)'', Preprint, \url{arXiv:1611.06827}].