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Tiling randomly perturbed multipartite graphs

Enrique Gomez-Leos, Ryan R. Martin

TL;DR

The paper determines the threshold for a perfect $K_r$-tiling in a balanced $r$-partite graph after random perturbation by a $G_r(n,p)$ edge-dense random graph, showing the sharp threshold $p \ge C n^{-2/r}$ (whp) when the host graph has minimum partite-degree $\delta^{*}(G) \ge \alpha n$. The authors develop a multipartite regularity framework together with a linear programming approach to obtain a perfect fractional $S^*$-tiling of the Szemerédi reduction, then transform this into an actual tiling via random slicing and careful regularization of the clusters, leftovers, and star structures. They prove both the upper-bound (existence) and lower-bound (non-existence) parts, including an extremal construction demonstrating sharpness up to constants and a note that a linear minimum-degree is necessary to avoid polylog factors. The results extend Balogh–Treglown–Wagner’s bipartite tiling threshold to the multipartite setting and clarify how regularity-based methods can bridge fractional to integral tilings in this more complex environment, with implications for tiling problems in perturbed multipartite networks.

Abstract

A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of the graph $K_r$ in $G$ that covers all vertices of $G$. In this paper, we prove that the threshold for the existence of a perfect $K_{r}$-tiling of a randomly perturbed balanced $r$-partite graph on $rn$ vertices is $n^{-2/r}$. This result is a multipartite analog of a theorem of Balogh, Treglown, and Wagner and extends our previous result, which was limited to the bipartite setting.

Tiling randomly perturbed multipartite graphs

TL;DR

The paper determines the threshold for a perfect -tiling in a balanced -partite graph after random perturbation by a edge-dense random graph, showing the sharp threshold (whp) when the host graph has minimum partite-degree . The authors develop a multipartite regularity framework together with a linear programming approach to obtain a perfect fractional -tiling of the Szemerédi reduction, then transform this into an actual tiling via random slicing and careful regularization of the clusters, leftovers, and star structures. They prove both the upper-bound (existence) and lower-bound (non-existence) parts, including an extremal construction demonstrating sharpness up to constants and a note that a linear minimum-degree is necessary to avoid polylog factors. The results extend Balogh–Treglown–Wagner’s bipartite tiling threshold to the multipartite setting and clarify how regularity-based methods can bridge fractional to integral tilings in this more complex environment, with implications for tiling problems in perturbed multipartite networks.

Abstract

A perfect -tiling in a graph is a collection of vertex-disjoint copies of the graph in that covers all vertices of . In this paper, we prove that the threshold for the existence of a perfect -tiling of a randomly perturbed balanced -partite graph on vertices is . This result is a multipartite analog of a theorem of Balogh, Treglown, and Wagner and extends our previous result, which was limited to the bipartite setting.

Paper Structure

This paper contains 24 sections, 29 theorems, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organization
  3. Preliminaries
  4. Notation
  5. Concentration inequalities
  6. Epsilon-regular pairs
  7. Szemerédi's Regularity Lemma
  8. Random multipartite graphs
  9. Linear programming
  10. Extremal example
  11. Proof of the main theorem
  12. Applying the Regularity Lemma
  13. Tiling the Szemerédi graph
  14. Cleaning up the stars
  15. Tiling leftover vertices
  16. ...and 9 more sections

Key Result

Theorem 1

Let $r\geq 2$ and let $n \in \mathbb{N}$ be divisible by $r$. For every $\alpha >0$, there is a $c=c(\alpha, r)>0$ such that if $p\geq cn^{-2/r}$ and $G$ is an $n$-vertex graph with $\delta(G) \geq \alpha n$, then $G \cup G(n,p)$ contains a perfect $K_r$-tiling whp.

Figures (2)

  • Figure 1: An instance of an $S\in S_{t}(1,2)$ in $\tilde{G}_{Sz}$. The center cluster is $U_1=U_1(S)$ and is in $V_1$. The big leaves are $T_1, \ldots, T_t$ and are in $V_2$. In each of $V_3,\ldots,V_r$, there is a small leaf.
  • Figure 2: The partitioning of a pair $(U_1, T_k)$ into $t$ many pairs $(U_{1}, T_k)$ for $1\leq k\leq t$. The region $T_{k}'$ bounded by the solid lines is $1/s$ proportion of $T_k$. The region $T_{k}"$ is of size $1-1/s$ proportion of $T_k$. The random tiling $\mathcal{T}_3$ will leave uncovered the vertices $T_{k}"'$. The pair $\bigl(U_{1,k}, T_{k}' \cup T_{k}"'\bigr)$ is shown to be $(5\epsilon_3, \delta_{\ref{['lemma: super-regularization']}} /5)$-super-regular.

Theorems & Definitions (47)

  • Theorem 1: Balogh, Treglown, Wagner balogh2019tilings, Theorem 1.3
  • Definition 2
  • Definition 3
  • Theorem 4: Keevash and Mycroft keevash2015multipartite
  • Theorem 5
  • Lemma 6: Chebyshev Inequality
  • Lemma 7: Chernoff Bounds, see JLR, Section 2.1
  • Lemma 8: Janson's Inequality, see randomgraphsFK, Theorem 21.12
  • Corollary 9
  • Remark
  • ...and 37 more