A hypergraph analog of Dirac's Theorem for long cycles in 2-connected graphs

TL;DR

This paper extends Dirac's classical cycle length result from graphs to hypergraphs, establishing conditions under which long Berge cycles exist in 2-connected r-uniform hypergraphs.

Contribution

It introduces a hypergraph analog of Dirac's theorem, providing a minimum degree condition for the existence of long Berge cycles in 2-connected hypergraphs.

Findings

Proves a minimum degree condition for long Berge cycles in hypergraphs

Establishes the bound as tight for all relevant parameters

Generalizes a fundamental graph theory result to hypergraphs

Abstract

Dirac proved that each $n$-vertex $2$-connected graph with minimum degree at least $k$ contains a cycle of length at least $\min\{2k, n\}$. We consider a hypergraph version of this result. A Berge cycle in a hypergraph is an alternating sequence of distinct vertices and edges $v_1,e_2,v_2, \ldots, e_c, v_1$ such that $\{v_i,v_{i+1}\} \subseteq e_i$ for all $i$ (with indices taken modulo $c$). We prove that for $n \geq k \geq r+2 \geq 5$, every $2$-connected $r$-uniform $n$-vertex hypergraph with minimum degree at least ${k-1 \choose r-1} + 1$ has a Berge cycle of length at least $\min\{2k, n\}$. The bound is exact for all $k\geq r+2\geq 5$.