$\mathrm{\ell }$-covering $k$-hypergraphs are quasi-eulerian

An Euler tour in a hypergraph $H$ is a closed walk that traverses each edge of $H$ exactly once, and an Euler family is a family of closed walks that jointly traverse each edge of $H$ exactly once. An $\mathrm{\ell }$-covering $k$-hypergraph, for $2\le \mathrm{\ell }<k$, is a $k$-uniform hypergraph in which every $\mathrm{\ell }$-subset of vertices lie together in at least one edge. In this paper we prove that every $\mathrm{\ell }$-covering $k$-hypergraph, for $k \ge 3$, admits an Euler family.