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Transversal tilings in k-partite graphs without large holes

Xinyu He, Xiangxiang Nie, Donglei Yang

TL;DR

The paper addresses transversal tilings in multipartite graphs that are subgraphs of the $n$-blow-up of $K_k$ or $C_k$, under sublinear hole constraints ${ m ho}^*_r(G) < eta n$. It develops a lattice-based absorbing framework that, together with a regularity-based almost-tiling step, yields transversal $K_k$-factors when ${ m abla}^*(G) \, ext{is}\, ( frac12+ ext{const})n$ with $r=k-1$, or transversal $C_k$-factors when ${ m abla}^*(G) \, ext{is}\, ( frac{2}{k}+ ext{const})n$ with ${ m ho}^*_2(G)=o(n)$. The approach achieves asymptotically tight thresholds for $k=3$ and extends prior results by incorporating $(k-1)$- and $k$-partite hole constraints, as well as providing a unified absorbing framework that can handle both cliques and cycles. The results advance our understanding of tiling in multipartite pseudo-random graphs and offer robust methods for finding spanning transversal factors in dense, structured settings. The techniques have potential implications for Ramsey–Turán type problems with independence constraints and for broader tiling questions in blown-up multipartite graphs.

Abstract

We show that for any constant $μ>0$ and $k\ge 3$, there exists $α>0$ such that the following holds for sufficiently large $n \in \mathbb{N}$. If $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $K_{k}$ with ${δ^*}(G)\geq (\frac{1}{2}+μ) n$ and $α^*_{k-1}(G)<αn$, then $G$ has a transversal $K_{k}$-factor. Moreover, the bound $\frac{1}{2}$ is asymptotically tight for the case \(k=3\). In addition, we show that if $k\ge 4$, $G=(V_{1},\ldots,V_{k},E)$ is a spanning subgraph of the $n$-blow-up of $C_{k}$ with ${δ^*}(G)\ge (\frac{2}{k}+μ) n$, and $α^*_{2}(G)<αn$, then $G$ has a transversal $C_{k}$-factor. This extends a recent result of Han, Hu, Ping, Wang, Wang and Yang.

Transversal tilings in k-partite graphs without large holes

TL;DR

The paper addresses transversal tilings in multipartite graphs that are subgraphs of the -blow-up of or , under sublinear hole constraints . It develops a lattice-based absorbing framework that, together with a regularity-based almost-tiling step, yields transversal -factors when with , or transversal -factors when with . The approach achieves asymptotically tight thresholds for and extends prior results by incorporating - and -partite hole constraints, as well as providing a unified absorbing framework that can handle both cliques and cycles. The results advance our understanding of tiling in multipartite pseudo-random graphs and offer robust methods for finding spanning transversal factors in dense, structured settings. The techniques have potential implications for Ramsey–Turán type problems with independence constraints and for broader tiling questions in blown-up multipartite graphs.

Abstract

We show that for any constant and , there exists such that the following holds for sufficiently large . If is a spanning subgraph of the -blow-up of with and , then has a transversal -factor. Moreover, the bound is asymptotically tight for the case . In addition, we show that if , is a spanning subgraph of the -blow-up of with , and , then has a transversal -factor. This extends a recent result of Han, Hu, Ping, Wang, Wang and Yang.
Paper Structure (11 sections, 13 theorems, 13 equations, 10 figures)

This paper contains 11 sections, 13 theorems, 13 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Notation.
  3. Organization.
  4. Proofs of Theorem \ref{['thm1']} and \ref{['thm2']}
  5. Absorbing sets
  6. obtaining absorbers
  7. Proof of Lemma \ref{['3']}
  8. Proof of Lemma \ref{['7']}
  9. Almost cover
  10. Almost perfect tilings
  11. Proof of Fact \ref{['0485']}

Key Result

Theorem 1.1

For every $k$, there exists $n_0 := n_0(k)$ such that whenever $n \ge n_0$ the following holds for every spanning subgraph $G$ of the $n$-blow-up of $K_k$ with $\delta^*(G) \ge \left(1 - \frac{1}{k}\right)n.$ The graph $G$ does not contain a $K_k$-factor if and only if both $n$ and $k$ are odd, $k$

Figures (10)

  • Figure 1: $P^*=u_2u_3u_4\cup\{u_1,u_5,\cdots, u_k\}$
  • Figure 2: $M=u_2u_3\cup u_4u_5\cup\{u_1,u_6,u_7\ldots,u_k\}$
  • Figure 3: $M=u_2u_3\cup u_4u_5\cup\{u_1,u_6,u_7\ldots,u_k\}$
  • Figure 4: $M=u_2u_3\cup u_4u_5\cup\{u_1,u_6,u_7\ldots,u_k\}$
  • Figure 5: $M'=u_2u_3\cup u_5u_6\cup\{u_1,u_4,u_7\ldots,u_k\}$
  • ...and 5 more figures

Theorems & Definitions (61)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 51 more