Hyunwoo Lee (이현우), 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 $G.$ Let ${\delta }_{sub}(n,H)$ 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 ${\delta }_{sub}(n,H)$. More precisely, for every graph $H$ with at least one edge, there is a constant $1<{\xi }^{\ast }(H)\le 2$ such that ${\delta }_{sub}(n,H)=(1-\frac{1}{{\xi }^{\ast }(H)}+o(1))n$ if $H$ has a bipartite subdivision with two parts having different parities. Otherwise, the threshold depends on the parity of $n$.