Using edge cuts to find Euler tours and Euler families in hypergraphs

TL;DR

This paper introduces new techniques and algorithms for determining the existence of Euler tours and families in hypergraphs by reducing the problem to smaller hypergraphs via edge cuts, with practical construction methods.

Contribution

It presents novel methods using edge cut assignments and collapsed hypergraphs to analyze Euler tours and families in hypergraphs, along with algorithms for their detection and construction.

Findings

Reduction of Euler tour existence to smaller hypergraphs

Introduction of edge cut assignment and collapsed hypergraph techniques

Algorithms for detecting and constructing Euler tours and families

Abstract

An Euler tour in a hypergraph is a closed walk that traverses each edge of the hypergraph exactly once, while an Euler family is a family of closed walks that jointly traverse each edge exactly once and cannot be concatenated. In this paper, we show how the problem of existence of an Euler tour (family) in a hypergraph $H$ can be reduced to the analogous problem in some smaller hypergraphs that are derived from $H$ using an edge cut of $H$. In the process, new techniques of edge cut assignments and collapsed hypergraphs are introduced. Moreover, we describe algorithms based on these characterizations that determine whether or not a hypergraph admits an Euler tour (family), and can also construct an Euler tour (family) if it exists.