Nested cycles with no geometric crossings

TL;DR

This paper addresses a 1975 question by Erdős, establishing the minimal number of edges needed in a graph to guarantee two nested, vertex-respecting, edge-disjoint cycles, proving the bound is linear in the number of vertices.

Contribution

The authors prove the optimal linear bound for Erdős's nested cycles problem using sublinear expanders, advancing understanding of cycle structures in graphs.

Findings

Established the linear bound f(n)=O(n) for nested cycles

Used sublinear expanders to prove the bound

Solved a longstanding problem posed by Erdős in 1975

Abstract

In 1975, Erd\H{o}s asked the following question: what is the smallest function $f(n)$ for which all graphs with $n$ vertices and $f(n)$ edges contain two edge-disjoint cycles $C_1$ and $C_2$, such that the vertex set of $C_2$ is a subset of the vertex set of $C_1$ and their cyclic orderings of the vertices respect each other? We prove the optimal linear bound $f(n)=O(n)$ using sublinear expanders.