Eulerian Properties of Design Hypergraphs and Hypergraphs with Small Edge Cuts

Abstract

An Euler tour of a hypergraph is a closed walk that traverses every edge exactly once; if a hypergraph admits such a walk, then it is called eulerian. Although this notion is one of the progenitors of graph theory --- dating back to the eighteenth century --- treatment of this subject has only begun on hypergraphs in the last decade. Other authors have produced results about rank-2 universal cycles and 1-overlap cycles, which are equivalent to our definition of Euler tours. In contrast, an Euler family is a collection of nontrivial closed walks that jointly traverse every edge of the hypergraph exactly once and cannot be concatenated simply. Since an Euler tour is an Euler family comprising a single walk, having an Euler family is a weaker attribute than being eulerian; we call a hypergraph quasi-eulerian if it admits an Euler family. A triple system is a 3-uniform hypergraph in which every pair of vertices lie in exactly $\lambda$ edges together, for some parameter $\lambda$. We first prove that all triple systems with at least two edges are eulerian. We then define a new kind of hypergraph, which we call $\ell$-covering $k$-hypergraphs: these are $k$-uniform hypergraphs in which every $\ell$-subset of the vertices lie in at least one edge together. We are able to show that such hypergraphs are eulerian when $\ell=k-1$ and quasi-eulerian otherwise, as long as they have at least two edges. Finally, we give some constructive results on hypergraphs with small edge cuts. There has been analogous work by other authors on hypergraphs with small vertex cuts. We reduce the problem of finding an Euler tour in a hypergraph to finding an Euler tour in each of the connected components of the edge-deleted subhypergraph, then show how these individual Euler tours can be concatenated.