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arXiv · 2605.17218

Induced subdivisions in graphs of large girth

Peiru Kuang·Yan Wang
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Abstract

In this paper, we prove that there exists an absolute constant $g_0$ such that, for every integer $k\ge 3$, every graph $G$ with $\delta(G)\ge k$ and $g(G)\ge g_0$ contains an induced subdivision of $K_{k+1}$. This fully resolves a problem raised by K\"{u}hn and Osthus (originally attributed to Shi), and improves a recent result of Gir\~{a}o and Hunter. Our proof uses some ideas from Gir\~{a}o and Hunter. Another main ingredient in our proof is an induced variant of Mader's theorem: for every fixed \(s,\eta,D\), every graph \(J\) with \(\Delta(J)\le D\), \(d(J)>s-2+\eta\) and sufficiently large girth contains an induced subdivision of \(K_s\).

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Peiru Kuang, Yan Wang. 2026-05-17. Induced subdivisions in graphs of large girth. https://arxiv.org/abs/2605.17218

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