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

Alexandr Kostochka, Ruth Luo, Grace McCourt

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 $k\geq r+2\geq 5$.

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

Abstract

Dirac proved that each -vertex -connected graph with minimum degree at least contains a cycle of length at least . We consider a hypergraph version of this result. A Berge cycle in a hypergraph is an alternating sequence of distinct vertices and edges such that for all (with indices taken modulo ). We prove that for , every -connected -uniform -vertex hypergraph with minimum degree at least has a Berge cycle of length at least . The bound is exact for all .
Paper Structure (16 sections, 19 theorems, 83 equations, 2 figures)

This paper contains 16 sections, 19 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction and Results
  2. Terminology and known results for graphs
  3. Terminology and known results for uniform hypergraphs
  4. Our results and structure of the paper
  5. Outline of the paper
  6. Short proofs
  7. Proof of Proposition \ref{['k=2!']}
  8. Proof of Corollary \ref{['kpath']} on codiameters
  9. Setup and simple properties of best lollipops
  10. Notation and setup
  11. Simple properties of best lollipops
  12. Nontrivial paths in best lollipops
  13. Vertex $u_{\ell+1}$ has few neighbors in $P$
  14. Partial Berge paths in best $p$-lollipops are long
  15. The paths in lollipops are short
  16. ...and 1 more sections

Key Result

Theorem 1

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

Figures (2)

  • Figure 1: An $o$-lollipop and a $p$-lollipop.
  • Figure 2: An o-lollipop $(C,P)$ with edge $f_0$ containing $u_{\ell+1}$.

Theorems & Definitions (66)

  • Theorem 1: Dirac D
  • Theorem 2: Dirac D
  • Theorem 3: Jackson jackson2
  • Definition 1.1
  • Theorem 4: Theorem 1.7 in KLM
  • Theorem 5: KLM
  • Definition 1.2
  • Definition 1.3
  • Corollary 6
  • Theorem 7
  • ...and 56 more