Cycles with almost linearly many chords

Abstract

We prove that constant minimum degree already forces cycles with almost linearly many chords. Specifically, every graph $G$ with $\delta(G)\ge C$ contains a cycle of length $\ell\ge 4$ with $\Omega(\ell/\log^{C}\ell)$ 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--Erd\H{o}s--Staton conjecture. It also gives a strong affirmative conclusion in the direction of a recent question of Dvo\v{r}\'ak, Martins, Thomass\'e, and Trotignon asking whether constant-degree graphs must contain cycles whose chord counts grow with their length.