On perfect subdivision tilings

For a given graph $H$, we say that a graph $G$ has a perfect $H$-subdivision tiling if $G$ contains a collection of vertex-disjoint subdivisions of $H$ covering all vertices of $G\mathrm{.}$ Let ${\delta }_{\mathrm{s}\mathrm{u}\mathrm{b}}(n,H)$ be the smallest integer $k$ such that any $n$-vertex graph $G$ with minimum degree at least $k$ has a perfect $H$-subdivision tiling. For every graph $H$, we asymptotically determined the value of ${\delta }_{\mathrm{s}\mathrm{u}\mathrm{b}}(n,H)$. More precisely, for every graph $H$ with at least one edge, there is an integer ${\mathrm{h}\mathrm{c}\mathrm{f}}_{\xi }(H)$ and a constant $1<{\xi }^{\ast }(H)\le 2$ that can be explicitly determined by structural properties of $H$ such that ${\delta }_{\mathrm{s}\mathrm{u}\mathrm{b}}(n,H)=(1-\frac{1}{{\xi }^{\ast }(H)}+o(1))n$ holds for all $n$ and $H$ unless ${\mathrm{h}\mathrm{c}\mathrm{f}}_{\xi }(H)=2$ and $n$ is odd. When ${\mathrm{h}\mathrm{c}\mathrm{f}}_{\xi }(H)=2$ and $n$ is odd, then we show that ${\delta }_{\mathrm{s}\mathrm{u}\mathrm{b}}(n,H)=(\frac{1}{2}+o(1))n$.