Transversal tilings in k-partite graphs without large holes

TL;DR

This paper establishes conditions under which large, structured k-partite graphs contain transversal complete or cycle factors, extending known bounds and demonstrating asymptotic tightness for certain cases.

Contribution

It provides new minimum degree and independence number conditions ensuring the existence of transversal factors in k-partite graphs, extending previous results and identifying tight bounds.

Findings

For k=3, the 1/2 bound is asymptotically tight.

Conditions guarantee transversal K_k and C_k factors in large k-partite graphs.

Extends recent results on cycle factors in blow-up graphs.

Abstract

We show that for any constant $\mu>0$ and $k\ge 3$, there exists $\alpha>0$ 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 ${\delta^*}(G)\geq (\frac{1}{2}+\mu) n$ and $\alpha^*_{k-1}(G)<\alpha n$, 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 ${\delta^*}(G)\ge (\frac{2}{k}+\mu) n$, and $\alpha^*_{2}(G)<\alpha n$, then $G$ has a transversal $C_{k}$-factor. This extends a recent result of Han, Hu, Ping, Wang, Wang and Yang.