Polylogarithmic Bounds for Nested Cycles without Geometric Crossings

TL;DR

This paper establishes polylogarithmic bounds for the existence of multiple nested, noncrossing cycles in large graphs, extending previous results that required higher average degrees.

Contribution

It introduces a polylogarithmic bound for nested cycles without crossings, improving upon prior constant degree conditions, using advanced expansion and layering techniques.

Findings

Graphs with sufficiently many edges contain k nested cycles without crossings.

The bound is polylogarithmic in the number of vertices, specifically involving (log n)^{k-1} and (log log n)^{k-3}.

The proof combines expander frameworks with a controlled wrapping lemma.

Abstract

A problem of Erd\H{o}s asks for extremal conditions forcing edge-disjoint cycles with a prescribed nested structure. In the geometric version, the nesting is required to be noncrossing with respect to the cyclic orders. Fern\'andez, Kim, Kim and Liu proved that constant average degree forces two such cycles. We prove a polylogarithmic bound for the natural multi-layer version: for every fixed $k\ge 3$, every sufficiently large $n$-vertex graph with at least \[ C_k n(\log n)^{k-1}(\log\log n)^{k-3} \] edges contains $k$ pairwise edge-disjoint nested cycles without geometric crossings. The proof combines the robust sublinear expander framework of Alon, Buci\'c, Sauermann, Zakharov and Zamir with a controlled wrapping lemma that permits the layers to be built successively with controlled length.