Counting odd cycles in locally dense graphs

TL;DR

This paper proves that sufficiently large locally dense graphs contain a number of odd cycles proportional to their size, addressing a question about cycle counts in such graphs.

Contribution

It establishes a lower bound on the number of odd cycles in locally dense graphs, advancing understanding of cycle enumeration in dense graph classes.

Findings

Large locally dense graphs contain at least (d^r - ε)|V(G)|^r odd cycles.

The number of cycles matches the expected count in random graphs with similar density.

Addresses a specific open question in graph theory about cycle counts.

Abstract

We prove that for any given $\varepsilon>0$ and $d\in [0,1]$, every sufficiently large $(\varepsilon, d)$-dense graph $G$ contains for each odd integer $r$ at least $(d^r-\varepsilon)|V(G)|^r$ cycles of length $r$. Here, $G$ being $(\varepsilon, d)$-dense means that every set $X$ containing at least~$\varepsilon\,|V(G)|$ vertices spans at least $\tfrac d2\, |X|^2$ edges, and what we really count is the number of homomorphisms from an $r$-cycle into $G$. The result adresses a question of Y. Kohayakawa, B. Nagle, V. R\"odl, and M. Schacht.