On Perfect Subdivision Tilings

Abstract

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 $1 < \xi^*(H)\leq 2$ such that $\delta_{sub}(n, H) = \left(1 - \frac{1}{\xi^*(H)} + o(1) \right)n$ if $H$ has a bipartite subdivision with two parts having different parities. Otherwise, the threshold may depend on the parity of $n.$