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Pancyclicity in hypergraphs with large uniformity

Teegan Bailey, Isaiah Hollars, Yupei Li, Ruth Luo

TL;DR

The paper extends Bondy–Dirac-type pancyclicity from graphs to $n$-vertex, $r$-uniform hypergraphs through Berge cycles, establishing exact minimum degree conditions for having Berge cycles of all lengths from $2$ to $n$ in the regime $r \\ge \\lfloor (n-1)/2\\rfloor-1$ when $n$ is large. The authors employ a Hamiltonian Berge cycle $C$ as a backbone, analyze the remaining edges, and develop shifting techniques $S_s$ and self-shift complementary (SSC) structures to generate chords that yield cycles of all lengths; they connect to existing graph results by passing to the incidence graph and invoking results of Brandt and Hu–Sun, culminating in two main theorems (mainbig and mainsmall) that cover the large-uniformity and small-regime cases. Their contributions include sharp Dirac-type pancyclicity bounds for all $3\le r \le n$ in the large-$n$ limit, accompanied by sharp constructions demonstrating tightness and a deeper toolkit (shifting and SSC) potentially applicable to related extremal hypergraph problems. Overall, the work broadens the scope of pancyclicity to hypergraphs, linking hypergraph cycles to classical graph theory via incidence structures and delivering a near-complete picture for large $n$ across the full range of uniformities.

Abstract

A Berge cycle of length $\ell$ in a hypergraph $\mathcal{H}$ is a sequence of alternating vertices and edges $v_0e_0v_1e_1...v_\ell e_\ell v_0$ such that $\{v_i,v_{i+1}\}\subseteq e_i$ for all $i$, with indices taken modulo $\ell$. For $n$ sufficiently large and $r\geq \lfloor\frac{n-1}{2}\rfloor-1$ we prove exact minimum degree conditions for an $n$-vertex, $r$-uniform hypergraph to contain Berge cycles of every length between $2$ and $n$. In conjunction with previous work, this provides sharp Dirac-type conditions for pancyclicity in $r$-uniform hypergraphs for all $3\leq r\leq n$ when $n$ is sufficiently large.

Pancyclicity in hypergraphs with large uniformity

TL;DR

The paper extends Bondy–Dirac-type pancyclicity from graphs to -vertex, -uniform hypergraphs through Berge cycles, establishing exact minimum degree conditions for having Berge cycles of all lengths from to in the regime when is large. The authors employ a Hamiltonian Berge cycle as a backbone, analyze the remaining edges, and develop shifting techniques and self-shift complementary (SSC) structures to generate chords that yield cycles of all lengths; they connect to existing graph results by passing to the incidence graph and invoking results of Brandt and Hu–Sun, culminating in two main theorems (mainbig and mainsmall) that cover the large-uniformity and small-regime cases. Their contributions include sharp Dirac-type pancyclicity bounds for all in the large- limit, accompanied by sharp constructions demonstrating tightness and a deeper toolkit (shifting and SSC) potentially applicable to related extremal hypergraph problems. Overall, the work broadens the scope of pancyclicity to hypergraphs, linking hypergraph cycles to classical graph theory via incidence structures and delivering a near-complete picture for large across the full range of uniformities.

Abstract

A Berge cycle of length in a hypergraph is a sequence of alternating vertices and edges such that for all , with indices taken modulo . For sufficiently large and we prove exact minimum degree conditions for an -vertex, -uniform hypergraph to contain Berge cycles of every length between and . In conjunction with previous work, this provides sharp Dirac-type conditions for pancyclicity in -uniform hypergraphs for all when is sufficiently large.
Paper Structure (7 sections, 12 theorems, 32 equations, 3 figures)

This paper contains 7 sections, 12 theorems, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof outline and tools
  3. Shifting functions
  4. Self-Shift Complementary Sets
  5. Proof of Theorem \ref{['mainbig']}
  6. Proof of Theorem \ref{['mainsmall']}
  7. Concluding remarks

Key Result

Theorem 1

Let $n \geq 3$. If $G$ is an $n$-vertex graph with minimum degree $\delta(G) \geq n/2$, then $G$ is hamiltonian.

Figures (3)

  • Figure 1: An $(n-s+1)$-Berge cycle in cases 1 and 2 respectively.
  • Figure 2: The set $\hat{O}_0$ with $n = 20, k=6$. Gray points are in $A$ and white points in $\overline{A}$.
  • Figure 3: Even length Berge cycles.

Theorems & Definitions (27)

  • Theorem 1: Dirac Dirac
  • Theorem 2: Bondy Bondy
  • Theorem 3: Kostochka, Luo, McCourt KLM
  • Theorem 4: Bailey, Li, Luo BLL
  • Theorem 5
  • Theorem 6: Brandt Brandt
  • Theorem 7: Hu, Sun HuSun
  • Theorem 8: Hu, Sun HuSun
  • Theorem 9
  • Theorem 10
  • ...and 17 more