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Pósa rotation through a random permutation

Nemanja Draganić, Peter Keevash

TL;DR

This work resolves the asymptotically optimal minimum-degree threshold for Hamiltonicity of a deterministic $n$-vertex graph $G$ perturbed by a random $2$-factor $F\sim G_{n,2}$, showing whp $G\\cup F$ is Hamiltonian when $\\delta(G)\\ge(1+o(1))\\sqrt{n\\log n/2}$ and this bound is tight up to lower-order terms. The authors combine random-permutation theory with a randomized Pósa rotation argument under multi-exposure to track how long and short cycles in $F$ can be connected through $G$. They also prove a stronger result for (approximately) regular graphs, where a polylogarithmic growth of the minimum degree suffices, using a variant based on special vertex sequences. Together, these results advance the theory of randomly perturbed graphs and open avenues for Hamiltonicity in sparser, structured bases and for other random-factor perturbations, including $C_\\ell$-factors.

Abstract

What minimum degree of a graph $G$ on $n$ vertices guarantees that the union of $G$ and a random $2$-factor (or permutation) is with high probability Hamiltonian? Girão and Espuny D{\'ı}az showed that the answer lies in the interval $[\tfrac15 \log n, n^{3/4+o(1)}]$. We improve both the upper and lower bounds to resolve this problem asymptotically, showing that the answer is $(1+o(1))\sqrt{n\log n/2}$. Furthermore, if $G$ is assumed to be (nearly) regular then we obtain the much stronger bound that any degree growing at least polylogarithmically in $n$ is sufficient for Hamiltonicity. Our proofs use some insights from the rich theory of random permutations and a randomised version of the classical technique of Pósa rotation adapted to multiple exposure arguments.

Pósa rotation through a random permutation

TL;DR

This work resolves the asymptotically optimal minimum-degree threshold for Hamiltonicity of a deterministic -vertex graph perturbed by a random -factor , showing whp is Hamiltonian when and this bound is tight up to lower-order terms. The authors combine random-permutation theory with a randomized Pósa rotation argument under multi-exposure to track how long and short cycles in can be connected through . They also prove a stronger result for (approximately) regular graphs, where a polylogarithmic growth of the minimum degree suffices, using a variant based on special vertex sequences. Together, these results advance the theory of randomly perturbed graphs and open avenues for Hamiltonicity in sparser, structured bases and for other random-factor perturbations, including -factors.

Abstract

What minimum degree of a graph on vertices guarantees that the union of and a random -factor (or permutation) is with high probability Hamiltonian? Girão and Espuny D{\'ı}az showed that the answer lies in the interval . We improve both the upper and lower bounds to resolve this problem asymptotically, showing that the answer is . Furthermore, if is assumed to be (nearly) regular then we obtain the much stronger bound that any degree growing at least polylogarithmically in is sufficient for Hamiltonicity. Our proofs use some insights from the rich theory of random permutations and a randomised version of the classical technique of Pósa rotation adapted to multiple exposure arguments.

Paper Structure

This paper contains 6 sections, 12 theorems, 4 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Random permutations
  3. Some applications of concentration
  4. Asymptotically optimal minimum degree
  5. Approximately regular graphs
  6. Concluding remarks

Key Result

Theorem 1.1

For any $\varepsilon>0$ there is $n_0$ so that the following holds for all $n>n_0$. Suppose $G$ is an $n$-vertex graph with $\delta(G)\geq (1+\varepsilon)\sqrt{n\log n/2}$. Let $F\sim G_{n,2}$ be a random $2$-regular graph on the same vertex set as $G$. Then whp $G\cup F$ is Hamiltonian.

Figures (3)

  • Figure 1: Using a good pair to incorporate a short cycle $C$ in the current path $P$.
  • Figure 2: A sequence $v_1 < \ldots < v_5$ of special vertices.
  • Figure 3: Incorporating the cycle $C_2$ into the long path on $V(C_1)$. The new path starts at $v_2^-$, following the blue and green paths and then the intervals $I_2,I_3,\ldots, I_k$. In the next step, $v_2^-$ plays the role of $v_1$ and we find the special vertices in $I_2$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • proof : Proof of \ref{['thm:lower bound minimum degree']}
  • Theorem 3.1: Chernoff bounds
  • Lemma 3.2
  • proof
  • Theorem 3.3: Bounded differences inequality
  • ...and 11 more