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 $G=(V_1,\dots ,V_k,E)$ is a spanning subgraph of the $n$-blow-up of $K_k$ with ${\delta }^{\ast }(G)\ge (\frac{1}{2}+\mu )n$ and ${\alpha }_{k-1}^{\ast }(G)<\alpha n$, 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 $k\ge 4$, $G=(V_1,\dots ,V_k,E)$ is a spanning subgraph of the $n$-blow-up of $C_k$ with ${\delta }^{\ast }(G)\ge (\frac{2}{k}+\mu )n$, and ${\alpha }_2^{\ast }(G)<\alpha n$, then $G$ has a transversal $C_k$-factor. This extends a recent result of Han, Hu, Ping, Wang, Wang and Yang.