Euler tours in hypergraphs

TL;DR

This paper proves that quasirandom hypergraphs with certain degree conditions contain tight Euler tours, confirming a longstanding conjecture about universal cycles for subsets.

Contribution

It establishes the existence of tight Euler tours in quasirandom hypergraphs under degree divisibility conditions, confirming a conjecture for complete hypergraphs.

Findings

Quasirandom hypergraphs have tight Euler tours when degrees are divisible by k.

Confirmed the 1989 conjecture on universal cycles for k-subsets.

Provided conditions under which Euler tours exist in hypergraphs.

Abstract

We show that a quasirandom $k$-uniform hypergraph $G$ has a tight Euler tour subject to the necessary condition that $k$ divides all vertex degrees. The case when $G$ is complete confirms a conjecture of Chung, Diaconis and Graham from 1989 on the existence of universal cycles for the $k$-subsets of an $n$-set.