The extremal function for cycles of length $\ell$ mod $k$

TL;DR

This paper characterizes the extremal function for cycles of length  mod k, linking it to the maximum average degree of certain cycle-free graphs and providing tight bounds for various cases.

Contribution

It establishes a proportional relationship between c_k() and the maximum average degree of C_-free graphs, refining bounds for cycle lengths mod k.

Findings

c_k() is proportional to the largest average degree of C_-free graphs on k vertices.

For even , c_k() = O( k^{2/}), tight for  4,6,10.

c_k() = \u2206() for  =     = )

Abstract

Burr and Erd\H{o}s conjectured that for each $k,\ell \in \mathbb Z^+$ such that $k \mathbb Z + \ell$ contains even integers, there exists $c_k(\ell)$ such that any graph of average degree at least $c_k(\ell)$ contains a cycle of length $\ell$ mod $k$. This conjecture was proved by Bollob\'{a}s, and many successive improvements of upper bounds on $c_k(\ell)$ appear in the literature. In this short note, for $1 \leq \ell \leq k$, we show that $c_k(\ell)$ is proportional to the largest average degree of a $C_{\ell}$-free graph on $k$ vertices, which determines $c_k(\ell)$ up to an absolute constant. In particular, using known results on Tur\'{a}n numbers for even cycles, we obtain $c_k(\ell) = O(\ell k^{2/\ell})$ for all even $\ell$, which is tight for $\ell \in \{4,6,10\}$. Since the complete bipartite graph $K_{\ell - 1,n - \ell + 1}$ has no cycle of length $2\ell$ mod $k$, it also shows $c_k(\ell) = \Theta(\ell)$ for $\ell = \Omega(\log k)$.