Not All Saturated 3-Forests Are Tight

TL;DR

This paper constructs a counterexample in hypergraph theory showing that not all saturated 3-forests are tight, thereby resolving an open problem and deepening understanding of hypergraph properties.

Contribution

It provides the first known example of a saturated 3-forest that is not tight, challenging previous assumptions in hypergraph theory.

Findings

Counterexample of a saturated 3-forest that is not tight

Resolves an open problem posed by Strausz

Enhances understanding of hypergraph properties

Abstract

A basic statement in graph theory is that every inclusion-maximal forest is connected, i.e. a tree. Using a definiton for higher dimensional forests by Graham and Lovasz and the connectivity-related notion of tightness for hypergraphs introduced by Arocha, Bracho and Neumann-Lara in, we provide an example of a saturated, i.e. inclusion-maximal 3-forest that is not tight. This resolves an open problem posed by Strausz.