[Paper Review] Essentially tight bounds for rainbow cycles in proper edge-colourings

Noga Alon, Matija Bucić+3 more|arXiv (Cornell University)|Sep 8, 2023

Limits and Structures in Graph TheoryMathematics3 citations

TL;DR

This paper establishes essentially tight bounds for the maximum average degree of a properly edge-coloured graph on $n$ vertices without a rainbow cycle, proving that $O(\log n \cdot \log \log n)$ average degree suffices to guarantee such a cycle. The result resolves a long-standing extremal problem by tightening the upper bound to within a $\log\log n$ factor of the known lower bound, and connects the problem to additive combinatorics in non-abelian groups.

ABSTRACT

An edge-coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on $n$ vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of $(\log n)^{2+o(1)}$ for this question. Very recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the $o(1)$ term in Tomon's bound, showing a bound of $O(\log^2 n)$. We prove an upper bound of $(\log n)^{1+o(1)}$ for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the $o(1)$ term, and so it essentially resolves this question. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non-abelian groups.

Motivation & Objective

• To determine the precise asymptotic threshold for the average degree in properly edge-coloured graphs that guarantees the existence of a rainbow cycle.
• To close the gap between the best-known lower bound of $\Omega(\log n)$ and the previously best upper bound of $O(\log^2 n)$.
• To establish that the $\log\log n$ factor in the upper bound is essentially necessary, resolving the extremal problem up to lower-order terms.
• To uncover and exploit connections between rainbow cycle problems and additive combinatorics in non-abelian groups, particularly the Davenport constant and transposition sets in symmetric groups.

Proposed method

• The authors use a novel structural analysis of properly edge-coloured graphs without rainbow cycles, focusing on the colour distribution and connectivity properties of subgraphs.
• They apply probabilistic and extremal techniques, including subsampling arguments and robust sublinear expander properties, to control the growth of colour-rich substructures.
• A key technique involves decomposing the colour palette into two parts and analyzing the connectivity of the induced subgraphs, leading to a refined bound on the average degree.
• The method leverages known results from additive combinatorics and extends them to non-abelian groups via reductions from the graph-theoretic problem.
• The proof combines graph-theoretic extremal arguments with tools from additive number theory, particularly the study of dissociated sets and the plus-minus Davenport constant.
• They use a reduction from the rainbow cycle problem to a problem on the symmetric group $\mathcal{S}_k$, analyzing the minimal size of a set of transpositions that can multiply to the identity.

Experimental results

Research questions

• RQ1 What is the maximum average degree of a properly edge-coloured $n$-vertex graph that avoids a rainbow cycle?
• RQ2 Can the $\log\log n$ factor in the upper bound $O(\log^2 n)$ be removed, or is it asymptotically necessary?
• RQ3 How are extremal bounds for rainbow cycles related to the additive Davenport constant in non-abelian groups?
• RQ4 What is the minimal size of a set of transpositions in $\mathcal{S}_k$ that can multiply to the identity in some order?
• RQ5 Can the colour palette of a robust sublinear expander be decomposed into two parts such that both parts induce connected subgraphs?

Key findings

• The paper establishes that any properly edge-coloured graph on $n$ vertices with average degree at least $C \cdot \log n \cdot \log \log n$ for some absolute constant $C > 0$ must contain a rainbow cycle.
• This bound is essentially tight, as the best-known lower bound for the threshold is $\Omega(\log n)$, and the $\log\log n$ factor is shown to be necessary up to constant factors.
• The result resolves the extremal problem of Keevash, Mubayi, Sudakov, and Verstra€«te by closing the gap between the best-known lower and upper bounds to within a $\log\log n$ factor.
• The authors establish a connection between the rainbow cycle problem and the plus-minus Davenport constant in non-abelian groups, showing that $d(n) \leq O(\log n \cdot \log \log n)$ for the additive dimension of any group of size $n$.
• For the symmetric group $\mathcal{S}_k$, the minimal size $t(k)$ of a set of transpositions that can multiply to the identity satisfies $k(\log k)^{1-o(1)} \leq t(k) \leq O(k(\log k)^2)$.
• The paper shows that the $\log\log n$ term in previous bounds is not an artifact of the method but likely necessary, and provides evidence that the bound is tight up to lower-order terms.