Minimum degree of 3-graphs without long linear paths

TL;DR

This paper extends Dirac's classical graph theory result to 3-uniform hypergraphs, establishing tight asymptotic lower bounds on minimum degree conditions needed to ensure long linear paths.

Contribution

It provides the first asymptotic bounds for minimum degrees in 3-graphs that guarantee linear paths of certain lengths, generalizing a fundamental graph theory theorem.

Findings

Derived asymptotic lower bounds for minimum degrees in 3-graphs

Bounds are tight up to a constant factor

Extended Dirac's theorem from graphs to hypergraphs

Abstract

A well known theorem in graph theory states that every graph $G$ on $n$ vertices and minimum degree at least $d$ contains a path of length at least $d$, and if $G$ is connected and $n\ge 2d+1$ then $G$ contains a path of length at least $2d$ (Dirac, 1952). In this article, we give an extension of Dirac's result to hypergraphs. We determine asymptotic lower bounds of the minimum degrees of 3-graphs to guarantee linear paths of specific lengths, and the lower bounds are tight up to a constant.