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Stability of transversal Hamilton cycles and paths

Yangyang Cheng, Katherine Staden

Abstract

Given graphs $G_1,\ldots,G_s$ all on a common vertex set and a graph $H$ with $e(H) = s$, a copy of $H$ is \emph{transversal} or \emph{rainbow} if it contains one edge from each $G_i$. We establish a stability result for transversal Hamilton cycles: the minimum degree required to guarantee a transversal Hamilton cycle can be lowered as long as the graph collection $G_1,\ldots,G_n$ is far in edit distance from several extremal cases. We obtain an analogous result for Hamilton paths. The proof is a combination of our newly developed regularity-blow-up method for transversals, along with the absorption method.

Stability of transversal Hamilton cycles and paths

Abstract

Given graphs all on a common vertex set and a graph with , a copy of is \emph{transversal} or \emph{rainbow} if it contains one edge from each . We establish a stability result for transversal Hamilton cycles: the minimum degree required to guarantee a transversal Hamilton cycle can be lowered as long as the graph collection is far in edit distance from several extremal cases. We obtain an analogous result for Hamilton paths. The proof is a combination of our newly developed regularity-blow-up method for transversals, along with the absorption method.
Paper Structure (16 sections, 22 theorems, 28 equations, 2 figures)

This paper contains 16 sections, 22 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General transversal embedding
  3. Transversal embedding of Hamilton cycles
  4. Stability for transversal Hamilton cycles and paths
  5. Notation and organisation
  6. Probabilistic tools
  7. Sketch of the proof of Theorems \ref{['th:rhc']} and \ref{['th:rhp']}
  8. The regularity-blow-up method for transversals
  9. Extremality and stability
  10. Extremal and stable graphs
  11. Stable graph collections
  12. Absorbing
  13. The stable case
  14. The extremal case
  15. Proofs of Theorems \ref{['th:rhc']} and \ref{['th:rhp']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

For all $\kappa>0$, there exist $\mu>0$ and $n_0$ such that the following holds for all integers $n \geq n_0$. Let $G$ be an $n$-vertex graph with $\delta(G) \geq (\frac{1}{2}-\mu)n$. If $G$ contains no Hamilton cycle, then either $G$ is $\kappa$-close to ; or $G$ is $\kappa$-close to with arbitrar

Figures (2)

  • Figure 1: A $c$-absorbing path of $(v,v)$ and a $c$-absorbing path of $(v,u)$ for $v \neq u$.
  • Figure 2: Finding a transversal Hamilton cycle in Lemma \ref{['lm:rhc']}(ii) and (iii). Here, $|A^{10}|-|A^{11}|=1$ and $|B^{10}|-|B^{11}|=0$. On the left, $|A^0|-|B^0|=0$, while on the right, $|A^0|-|B^0|=1$.

Theorems & Definitions (55)

  • Theorem 1.2: Stability for Dirac's theorem, folklore
  • Definition 1.3: $\bm{H}^b_a$, half-split graph collection
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 1.6
  • Definition 2.1: Regularity and superregularity
  • Lemma 2.2: Typical vertices and colours Cheng4
  • Lemma 2.3: Slicing lemma Cheng4
  • Lemma 2.4: Regularity lemma for graph collections Cheng4
  • Definition 2.5: Reduced graph collection
  • ...and 45 more