Cycles with almost linearly many chords
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866914252099420160 |
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| author | Draganić, Nemanja Girão, António |
| author_facet | Draganić, Nemanja Girão, António |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08769 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cycles with almost linearly many chords Draganić, Nemanja Girão, António Combinatorics 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. |
| title | Cycles with almost linearly many chords |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.08769 |