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Cycles with almost linearly many chords

Nemanja Draganić, António Girão

TL;DR

The paper proves that a constant minimum degree forces cycles with nearly linear numbers of chords: for some $C>0$, every graph with $\delta(G)\ge C$ contains a cycle of length $\ell$ with $\Omega(\ell/\log^{C}\ell)$ chords. The authors develop a gadget-based framework built from cycle extenders and nice spiders, operating within a robust sublinear expander backbone to convert local structure into globally chord-rich cycles. Central to the approach is a careful expander decomposition, a maximal gadget collection, and a strategic routing scheme that either yields many chords or leads to a contradiction under expansion and degree constraints. This yields a near-resolution to the Chen–Erdős–Staton conjecture and answers a question of Dvořák et al. by showing constant-degree graphs can contain cycles whose chord counts grow with length, illustrating the power of sublinear expansion in sparse graphs.

Abstract

We prove that constant minimum degree already forces cycles with almost linearly many chords. Specifically, every graph $G$ with $δ(G)\ge C$ contains a cycle of length $\ell\ge 4$ with $Ω(\ell/\log^{C}\ell)$ chords for some absolute constant $C>0$. This is the first result showing that a constant-degree condition yields an unbounded -- indeed nearly linear -- number of chords, placing our bound within a polylogarithmic factor of the Chen--Erdős--Staton conjecture. It also gives a strong affirmative conclusion in the direction of a recent question of Dvořák, Martins, Thomassé, and Trotignon asking whether constant-degree graphs must contain cycles whose chord counts grow with their length.

Cycles with almost linearly many chords

TL;DR

The paper proves that a constant minimum degree forces cycles with nearly linear numbers of chords: for some , every graph with contains a cycle of length with chords. The authors develop a gadget-based framework built from cycle extenders and nice spiders, operating within a robust sublinear expander backbone to convert local structure into globally chord-rich cycles. Central to the approach is a careful expander decomposition, a maximal gadget collection, and a strategic routing scheme that either yields many chords or leads to a contradiction under expansion and degree constraints. This yields a near-resolution to the Chen–Erdős–Staton conjecture and answers a question of Dvořák et al. by showing constant-degree graphs can contain cycles whose chord counts grow with length, illustrating the power of sublinear expansion in sparse graphs.

Abstract

We prove that constant minimum degree already forces cycles with almost linearly many chords. Specifically, every graph with contains a cycle of length with chords for some absolute constant . This is the first result showing that a constant-degree condition yields an unbounded -- indeed nearly linear -- number of chords, placing our bound within a polylogarithmic factor of the Chen--Erdős--Staton conjecture. It also gives a strong affirmative conclusion in the direction of a recent question of Dvořák, Martins, Thomassé, and Trotignon asking whether constant-degree graphs must contain cycles whose chord counts grow with their length.
Paper Structure (11 sections, 14 theorems, 9 equations, 2 figures)

This paper contains 11 sections, 14 theorems, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation.
  4. Sublinear expanders
  5. Cycle extenders
  6. Cycles with many chords
  7. Gadgets and how we use them
  8. Controling high degree vertices
  9. Maximal collection of gadgets and the structure outside
  10. Getting another gadget and completing the proof.
  11. Concluding remarks

Key Result

Theorem 1.1

There exists constants $c,C>0$ such that every graph $G$ with $\delta(G)\geqslant C$ contains a cycle of length $\ell$ with at least $\Omega\mathopen{}\mathclose{\left(\frac{\ell}{\log^{c}\ell}\right)$ chords, for some positive integer $\ell$.

Figures (2)

  • Figure 1: Depending on where the vertex-disjoint paths meet the first cycle (either on opposite sides of each chord or not), we obtain the two configurations shown. In both cases, the blue cycle contains a chord and includes at least half of the vertices of the second cycle.
  • Figure 2: The two types of useful structures: a nice spider on the left, and a cycle extender on the right

Theorems & Definitions (39)

  • Theorem 1.1
  • Proposition 2.1: Block--cut structure
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5: Sublinear expander
  • Theorem 2.6
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • ...and 29 more