Powers of Hamiltonian cycles in randomly augmented Pósa-Seymour graphs

TL;DR

This work analyzes how many random edges are needed to boost a Dirac-type graph with $\delta(G)\ge\left(\tfrac{k}{k+1}+\varepsilon\right)n$ into containing the $m$-th power of a Hamiltonian cycle. It introduces the $f_{k,m}$ family and the associated thresholds $\lambda_{k,m}$ and $\ell_{k,m}$ to bound the Dirac exponents $\eta_{k,m}$ and over-exponents $\bar{\eta}_{k,m}$, proving asymptotically tight two-sided bounds that become equal for infinitely many $m$ when $\lambda_{k,m}$ is integral (via Pell equations). The paper also provides explicit results for small $(k,m)$ (notably $k=2,3$) and develops the braid-based upper-bound machinery, together with a general 0-statement framework, to distinguish ordinary thresholds from over-thresholds. The results deepen understanding of threshold phenomena in randomly perturbed graphs and offer precise predictions and methods for when higher powers of Hamilton cycles appear under perturbations, with broader implications for Dirac-type problems in probabilistic settings.

Abstract

We study the question of the least number of random edges that need to be added to a Pósa-Seymour graph, that is, a graph with minimum degree exceeding $\frac k{k+1}n$, to secure the existence of the $m$-th power of a Hamiltonian cycle, $m>k$. It turns out that, depending on $k$ and $m$, this quantity may be captured by two types of thresholds, with one of them, called over-threshold, becoming dominant for large $m$. Indeed, for each $k\ge2$ and $m>m_0(k)$, we establish asymptotically tight lower and upper bounds on the over-thresholds (provided they exist) and show that for infinitely many instances of $m$ the two bounds coincide. In addition, we also determine the thresholds for some small values of $k$ and $m$.

Figures (6)

• Figure 2.1: The decomposition of an $8$-path.
• Figure 4.1: A 3-path on 12 vertices and a $V_0$-based partition with $V_0=\{1,2,8,11,12\}$ (the sets $T_0=\varnothing$ and $T_q=\varnothing$ not shown). The thick edges show the order in which the 3-path traverses through the vertex set.
• Figure 4.2: Illustration to step (1) of algorithm REWIRE. Operation $\textrm{SHIFT\_RIGHT}(S_i,S_{i+1},2)$ moves the last two vertices of $S_i$ (red) to $S_{i+1}$. Some of the original edges (dashed red) inside $S_i$ disappear. Instead, at least as many new edges (red) are added.
• Figure 4.3: Illustration to step (2) of algorithm REWIRE. Operation $\textrm{SHIFT\_RIGHT}(T_{i},T_{i+1},1)$ moves the last vertex (red) of $T_{i}$ to $T_{i+1}$. Some of the original bridge-edges (dashed red) between $S_{i+1}$ and $S_{i+2}$ disappear, and no new edges are created.
• Figure 4.4: Illustration to step (3) of algorithm REWIRE. Operation $\textrm{SHIFT\_LEFT}(S_i,S_{i+1},1)$ moves the first vertex (red) of $S_{i+1}$ to $S_{i}$. One of the original edges (dashed red) in $S_{i+1}$ disappears and no new edges are added.

Theorems & Definitions (30)

• Definition 1.1
• Theorem 1.2: ADRRS
• Definition 1.3
• Theorem 1.4: ADR
• Proposition 1.5
• Theorem 1.6
• Remark 1.7
• Proposition 1.8
• Definition 2.1
• Lemma 2.2: ADRRS, Prop. 5.8