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Pósa-type results for Berge-hypergraphs

Nika Salia

Abstract

A Berge cycle of length $k$ in a hypergraph $\mathcal H$ is a sequence of distinct vertices and hyperedges $v_1,h_1,v_2,h_2,\dots,v_{k},h_k$ such that $v_{i},v_{i+1}\in h_i$ for all $i\in[k]$, indices taken modulo $k$. Füredi, Kostochka and Luo recently gave sharp Dirac-type minimum degree conditions that force non-uniform hypergraphs to have Hamiltonian Berge cycles. We give a sharp Pósa-type lower bound for $r$-uniform and non-uniform hypergraphs that force Hamiltonian Berge cycles.

Pósa-type results for Berge-hypergraphs

Abstract

A Berge cycle of length in a hypergraph is a sequence of distinct vertices and hyperedges such that for all , indices taken modulo . Füredi, Kostochka and Luo recently gave sharp Dirac-type minimum degree conditions that force non-uniform hypergraphs to have Hamiltonian Berge cycles. We give a sharp Pósa-type lower bound for -uniform and non-uniform hypergraphs that force Hamiltonian Berge cycles.

Paper Structure

This paper contains 9 sections, 7 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Hamiltonian cycles in graphs
  3. Hamiltonicity for hypergraphs and main results
  4. Examples showing the sharpness of conditions
  5. Proofs
  6. Proof of the theorem for uniform hypergraphs
  7. Proof of the theorem for non-uniform hypergraphs
  8. Concluding remarks
  9. Acknowledgements

Key Result

Theorem 1

Let $G$ be an $n$ vertex graph. Let $n\geq 3$ and the degree sequence of $G$ be $d_1\leq d_2\leq \dots \leq d_n$. If for all $k<\frac{n}{2}$ the inequality $k<d_k$ holds then $G$ is Hamiltonian.

Theorems & Definitions (30)

  • Theorem 1: Pósa posa1962theorem
  • Theorem 2: Chvátal chvatal1972hamilton
  • Definition 3
  • Theorem 4: Füredi, Kostochka, Luo furedi2020berge
  • Theorem 5
  • Theorem 6
  • Example 1
  • Example 2
  • Example 3
  • Claim 7
  • ...and 20 more