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arXiv · 2609.02234

A linear bound for nested cycles without geometric crossings

Jiangdong Ai·Gregory Gutin·Yiming Hao
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Abstract

Cycles $C_1,\ldots,C_k$ in a graph are called nested without geometric crossings if they are pairwise edge-disjoint, $V(C_k)\subseteq\cdots\subseteq V(C_1)$, and each pair of consecutive cycles induces the same cyclic order on the vertices of the inner cycle, up to reversal. Let $f_k(n)$ be the least number of edges that forces such a family in every $n$-vertex graph. Gil Fern\'andez, Kim, Kim and Liu proved that $f_2(n)=O(n)$, answering a question of Erd\H{o}s, and asked whether $f_k(n)=O_k(n)$ for every fixed $k$. We prove this for all $k$. The proof selects the inner cycles together with a disjoint subgraph that supplies their external neighbours. A reselection argument gives disjoint paths from every inner-cycle vertex to any sufficiently large target set. Sublinear expansion and a rooted clique minor then allow the vertices to be joined in the required cyclic order.

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Jiangdong Ai, Gregory Gutin, Yiming Hao. 2026-09-02. A linear bound for nested cycles without geometric crossings. https://arxiv.org/abs/2609.02234

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