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The perturbation threshold of degenerate graphs

Jie Han, Seonghyuk Im, Bin Wang, Junxue Zhang

Abstract

We show that for any $d\ge 2$ and $Δ>0$ there exists $η>0$ such that the following holds: Let $G$ be an $n$-vertex graph with at least $Ω(n^2)$ edges and let $H$ be an $n$-vertex $d$-degenerate graph with maximum degree at most $Δ$. Then with high probability, $G \cup G(n, n^{-1/d - η})$ contains a copy of $H$. We also prove that the same conclusion extends to $d$-regular graphs with $d\ge 4$ satisfying a certain edge expansion property, with the threshold improved to $n^{-2/d - η}$. Such a property is satisfied by almost all $d$-regular graphs and for even $d$, by the $(d/2)$-th power of a Hamilton cycle.

The perturbation threshold of degenerate graphs

Abstract

We show that for any and there exists such that the following holds: Let be an -vertex graph with at least edges and let be an -vertex -degenerate graph with maximum degree at most . Then with high probability, contains a copy of . We also prove that the same conclusion extends to -regular graphs with satisfying a certain edge expansion property, with the threshold improved to . Such a property is satisfied by almost all -regular graphs and for even , by the -th power of a Hamilton cycle.
Paper Structure (10 sections, 10 theorems, 20 equations, 2 figures)

This paper contains 10 sections, 10 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Randomly perturbed graphs
  3. Notation and Preliminaries
  4. Basic notation
  5. Spreadness
  6. Proof of Theorem \ref{['combin']}
  7. Proofs of Theorems \ref{['maindegenerate']} and \ref{['mainthm:regular']}
  8. Proofs of Theorems \ref{['maindegenerate']} and \ref{['mainthm:regular']}
  9. Proofs of Corollaries \ref{['cor:almost_2dreg']} and \ref{['cor:Hcyclepower']}
  10. Concluding Remarks

Key Result

Theorem 1.1

For every integer $d\geq2$ and constants $\varepsilon, \Delta > 0$, there exists $\eta>0$ such that the following holds. If $G$ is an $n$-vertex graph with at least $\varepsilon n^2$ edges and $H$ is an $n$-vertex $d$-degenerate graph with maximum degree at most $\Delta$, then w.h.p. $G \cup G(n, n^

Figures (2)

  • Figure 1: $S(Q)$ where dotted arcs are edges of $M^-(Q)$
  • Figure 2: An example for $E=M^-(X)$, the blue edegs.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1: Spread
  • Proposition 2.2
  • Definition 2.3: Vertex-spread
  • Proposition 2.4
  • Lemma 3.1
  • ...and 8 more