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On perfect subdivision tilings

Part of: Graph theory

Published online by Cambridge University Press:  27 January 2025

Hyunwoo Lee
Hyunwoo Lee*
Affiliation:
Department of Mathematical Sciences, Institute for Basic Science (IBS), South Korea and Extremal Combinatorics and Probability Group(ECOPRO), KAIST, South Korea
*
Email: [email protected]
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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

Subdivisiontilingabsorption

MSC classification

Primary: 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , First View , pp. 1 - 24
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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