Transversal tilings in k-partite graphs without large holes

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.