The Extremal Function for Cycles of Length $\ell$ mod $k$

Benny Sudakov,Jacques Verstraete

doi:10.37236/6257

Abstract

Burr and Erdős 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 Bollobás, 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 Turán 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 $\ell = \Omega(\log k)$.

Highlights

More than forty years ago, Burr and Erdos conjectured that for each k, l ∈Z+ such that kZ + l contains even integers, there exists ck(l) such that any graph of average degree at least ck(l) contains a cycle of length l mod k
Thomassen [27] conjectured that every graph of minimum degree at least k + 1 contains cycles of all possible even lengths mod k
Since every graph with average degree a least 2k contains a subgraph of minimum degree at least k + 1, this conjecture implies that ck(l) ≤ 2k when k is even

Summary

Introduction

Z+ such that kZ + l contains even integers, there exists ck(l) such that any graph of average degree at least ck(l) contains a cycle of length l mod k. Bollobas [1] was the first to show that ck(l) exists, showing ck(l) ≤ This upper bound was reduced by a number of authors [3, 6, 7, 8, 9, 10, 12, 16, 27]. Thomassen [27] conjectured that every graph of minimum degree at least k + 1 contains cycles of all possible even lengths mod k. Liu and Ma [19] showed for k even that a graph of minimum degree at least k + 1 contains cycles of all possible even lengths mod k, answering a conjecture of Thomassen [27] for even values of k. Since any graph whose blocks are cliques of order k + 1 contains no cycle of length 2 modulo k, this is best possible

Main Theorem

Quantitative results

Remarks on chromatic number and cycles mod k

Preliminaries

Proof of Theorem 1

Concluding remarks