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On perfect subdivision tilings
Part of:
Graph theory
Published online by Cambridge University Press: 27 January 2025
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 
$1 \lt \xi ^*(H)\leq 2$ 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 
$\delta _{\mathrm {sub}}(n, H) = \left (\frac {1}{2} + o(1) \right )n$.
Keywords
MSC classification
Primary:
05C35: Extremal problems
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- © The Author(s), 2025. Published by Cambridge University Press
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