On perfect subdivision tilings
Hyunwoo Lee
TL;DR
This work characterizes the asymptotically exact minimum degree threshold for the existence of a perfect $H$-subdivision tiling in every $n$-vertex graph. It introduces a subdivision-aware framework built on the absorption method, the regularity lemma, and domination-based control of the reduced graph, centered around the parameters $\xi(H)$, $\mathrm{hcf}_{\xi}(H)$, and the derived constant $\xi^*(H)$. The main results show that $\delta_{\mathrm{sub}}(n,H) = \left(1 - \frac{1}{\xi^*(H)} + o(1)\right)n$ for all $H$ with at least one edge unless $\mathrm{hcf}_{\xi}(H)=2$, in which case parity of $n$ drives the threshold: $\delta_{\mathrm{sub}}(n,H) = \left(\tfrac{1}{2} + o(1)\right)n$ for odd $n$ and $\delta_{\mathrm{sub}}(n,H) = \left(1 - \frac{1}{\xi^*(H)} + o(1)\right)n$ for even $n$. As concrete corollaries, the results determine $\delta_{\mathrm{sub}}(n,K_r)$ for small $r$ and reveal non-monotonic behavior of subdivision tilings relative to ordinary tilings. The methods combine absorbers of three types, the regularity framework, and domination arguments to manage divisibility and space barriers, offering a versatile approach to subdivision-tiling problems with broad potential applications.
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 $δ_{\mathrm{sub}}(n, H)$ 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 $δ_{\mathrm{sub}}(n, H)$. More precisely, for every graph $H$ with at least one edge, there is an integer $\mathrm{hcf}_ξ(H)$ and a constant $1 < ξ^*(H)\leq 2$ that can be explicitly determined by structural properties of $H$ such that $δ_{\mathrm{sub}}(n, H) = \left(1 - \frac{1}{ξ^*(H)} + o(1) \right)n$ holds for all $n$ and $H$ unless $\mathrm{hcf}_ξ(H) = 2$ and $n$ is odd. When $\mathrm{hcf}_ξ(H) = 2$ and $n$ is odd, then we show that $δ_{\mathrm{sub}}(n, H) = \left(\frac{1}{2} + o(1) \right)n$.