Spanning Euler Tours in Hypergraphs

Abstract

Motivated by generalizations of de Bruijn cycles to various combinatorial structures (Chung, Diaconis, and Graham), we study various Euler tours in set systems. Let $\mathcal{G}$ be a hypergraph whose corank and rank are $c\ge 3$ and $k$, respetively. The minimum $t$-degree of $\mathcal{G}$ is the fewest number of edges containing every $t$-subset of vertices. An Euler tour (family, respectively) in $\mathcal{G}$ is a (family of, respectively) closed walk(s) that (jointly, respectively) traverses each edge of $\mathcal{G}$ exactly once. An Euler tour is spanning if it traverses all the vertices of $\mathcal{G}$. We show that $\mathcal{G}$ has an Euler family if its incidence graph is $(1+\lceil k/c\rceil )$-edge-connected. Provided that the number of vertices of $\mathcal{G}$ meets a reasonable lower bound, and either $2$-degree is at least $k$ or $t$-degree is at least one for $t\ge 3$, we show that $\mathcal{G}$ has a spanning Euler tour. To exhibit the usefulness of our results, we solve a number of open problems concerning ordering blocks of a design (these have applications in other fields such as erasure-correcting codes). Answering a question of Horan and Hurlbert, we show that a Steiner quadruple system of order $n$ has a (spanning) Euler tour if and only if $n\ge 8$ and $n\equiv 2,4(\mathrm{mod}6)$, and we prove a similar result for all Steiner systems, as well as all designs except for 2-designs whose index $\lambda$ is less than the largest block size. We nearly solve a conjecture of Dewar and Stevens on the existence of universal cycles in pairwise balanced designs. Motivated by R.L. Graham's question on the existence of Hamiltonian cycles in block-intersection graphs of Steiner triple systems, we establish the Hamiltonicity of the block-intersection graph of a large family of (not necessarily uniform) designs. All our results are constructive and of polynomial time complexity.