On perfect subdivision tilings

TL;DR

This paper determines the minimum degree threshold for perfect subdivisions of a fixed graph H in large graphs, revealing structural dependencies and asymptotic formulas with specific exceptions based on graph properties.

Contribution

It asymptotically characterizes the minimum degree needed for perfect H-subdivision tilings, introducing explicit structural parameters that influence the threshold.

Findings

Established asymptotic formulas for elta_sub(n, H)

Identified structural parameters hf_(H) and x^*(H) affecting thresholds

Described special case when hf_(H)=2 and n is odd

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 $\delta_{\mathrm{sub}}(n, H)$ 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 $\delta_{\mathrm{sub}}(n, H)$. More precisely, for every graph $H$ with at least one edge, there is an integer $\mathrm{hcf}_{\xi}(H)$ and a constant $1 < \xi^*(H)\leq 2$ that can be explicitly determined by structural properties of $H$ such that $\delta_{\mathrm{sub}}(n, H) = \left(1 - \frac{1}{\xi^*(H)} + o(1) \right)n$ holds for all $n$ and $H$ unless $\mathrm{hcf}_{\xi}(H) = 2$ and $n$ is odd. When $\mathrm{hcf}_{\xi}(H) = 2$ and $n$ is odd, then we show that $\delta_{\mathrm{sub}}(n, H) = \left(\frac{1}{2} + o(1) \right)n$.