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The Hamilton space of pseudorandom graphs

Micha Christoph, Rajko Nenadov, Kalina Petrova

Abstract

We show that if $n$ is odd and $p \ge C \log n / n$, then with high probability Hamilton cycles in $G(n,p)$ span its cycle space. More generally, we show this holds for a class of graphs satisfying certain natural pseudorandom properties. The proof is based on a novel idea of parity-switchers, which can be thought of as analogues of absorbers in the context of cycle spaces. As another application of our method, we show that Hamilton cycles in a near-Dirac graph $G$, that is, a graph $G$ with odd $n$ vertices and minimum degree $n/2 + C$ for sufficiently large constant $C$, span its cycle space.

The Hamilton space of pseudorandom graphs

Abstract

We show that if is odd and , then with high probability Hamilton cycles in span its cycle space. More generally, we show this holds for a class of graphs satisfying certain natural pseudorandom properties. The proof is based on a novel idea of parity-switchers, which can be thought of as analogues of absorbers in the context of cycle spaces. As another application of our method, we show that Hamilton cycles in a near-Dirac graph , that is, a graph with odd vertices and minimum degree for sufficiently large constant , span its cycle space.
Paper Structure (16 sections, 14 theorems, 39 equations, 1 figure)

This paper contains 16 sections, 14 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation.
  3. Proof Strategy
  4. Proof strategy.
  5. Case 1: $N_\mathrm{odd}^{\ell}(u) \cap N_\mathrm{even}^{\ell}(u) \neq \emptyset$.
  6. Case 2: $N_\mathrm{odd}^{\ell}(u) \cap N_\mathrm{even}^{\ell}(u) = \emptyset$.
  7. Step \ref{['strategy:cycle']}.
  8. Step \ref{['strategy:paths']}.
  9. Step \ref{['strategy:ham']}.
  10. Properties of pseudorandom graphs
  11. Connecting vertices
  12. Hamilton space in sparse pseudorandom graphs
  13. Step \ref{['strategy:cycle']}.
  14. Step \ref{['strategy:paths']}.
  15. Step \ref{['strategy:ham']}.
  16. ...and 1 more sections

Key Result

Theorem 1.1

There exists $C > 1$ such that the following holds for sufficiently large odd $n$. Let $G$ be a graph with $n$ vertices such that $\delta(G) \ge n/2 + C$. Then $\mathcal{C}_n(G) = \mathcal{C}(G)$.

Figures (1)

  • Figure 1: A Hamilton path with an even number of edges from $R$ (represented in red) in a parity-switcher.

Theorems & Definitions (32)

  • Theorem 1.1
  • Definition 1.2: THOMASON1987307
  • Theorem 1.3
  • Corollary 1.4
  • proof
  • Corollary 1.5
  • Corollary 1.6
  • proof
  • Lemma 2.1
  • proof
  • ...and 22 more