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arXiv · 2608.15786

Spanning $H$-subdivisions with Prescribed Path Lengths

Zhilan Wang·Shuo Wei·Jin Yan
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Abstract

We study spanning $H$-subdivisions in dense graphs where the length of every subdivision path is prescribed in advance. This problem is motivated in part by a question of Pavez-Sign\'e [Combin. Probab. Comput. 33 (2024), 121--128], who asked whether the subdivision paths in a spanning $H$-subdivision can be required to have similar lengths. Let $h\ge3$ be an integer and let $0<\beta\ll\alpha\ll1/h$. We prove that, for all sufficiently large $n$, every $n$-vertex graph $G$ with $\delta(G)\ge n/2+\lfloor h/3\rfloor$ has the following property. For every graph $H$ with $h$ edges and no isolated vertices, write $E(H)=\{e_1,\ldots,e_h\}$, and every choice of integers $\ell_1,\ldots,\ell_h\ge4$ satisfying $\sum_{i=1}^h\ell_i=n-|V(H)|+h$ and $\sum_{\ell_i<\alpha n}\ell_i\le\beta n$, the graph $G$ contains a spanning $H$-subdivision in which the $i$th edge of $H$ is replaced by a path of length exactly $\ell_i$. We also give a family of examples showing that a linear additive term in $h$ is necessary in general.

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BibTeXRIS

Zhilan Wang, Shuo Wei, Jin Yan. 2026-08-16. Spanning $H$-subdivisions with Prescribed Path Lengths. https://arxiv.org/abs/2608.15786

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