Towards Graham's rearrangement conjecture via rainbow paths

TL;DR

We study Graham's rearrangement conjecture in general groups by translating it to rainbow directed paths in edge-coloured Cayley graphs: a subset $S$ of a group $\\Gamma$ is rearrangeable iff $\\mathrm{Cay}(\\Gamma,S)$ contains a rainbow path of length $|S|-1$. We prove an asymptotic version: for any finite group $\\Gamma$ and subset $S$, there is a permutation of $S$ in which a $(1-o(1))|S|$-fraction of the partial products are distinct, yielding the first general asymptotic Graham-type result with no restrictions on $\\Gamma$ or $S$. The proof develops a unifying framework based on rainbow subgraphs, including a robust-expansion toolbox, dense-regime reductions, and mop-based arguments to handle general and abelian/non-abelian settings; Pollard's inequality is used to treat the $\\mathbb{Z}_p$ case. Together with asymptotic Schrijver-type results for rainbow paths in coloured graphs and their directed analogues, the work provides near-optimal rearrangements across broad classes of groups and highlights expansion phenomena as a unifying theme in combinatorial group theory.

Abstract

We study an old question in combinatorial group theory which can be traced back to a conjecture of Graham from 1971. Given a group $Γ$, and some subset $S\subseteq Γ$, is it possible to permute $S$ as $s_1, s_2, \ldots, s_d$ so that the partial products $\prod_{1 \leq i \leq t} s_i$, $t\in [d]$ are all distinct? Most of the progress towards this problem has been in the case when $Γ$ is a cyclic group. We show that for any group $Γ$ and any $S \subseteq Γ$, there is a permutation of $S$ where all but a vanishing proportion of the partial products are distinct, thereby establishing the first asymptotic version of Graham's conjecture under no restrictions on $Γ$ or $S$. To do so, we explore a natural connection between Graham's problem and the following very natural question attributed to Schrijver. Given a $d$-regular graph $G$ properly edge-coloured with $d$ colours, is it always possible to find a rainbow path with $d-1$ edges? We settle this question asymptotically by showing one can find a rainbow path of length $d - o(d)$. While this has immediate applications to Graham's question for example when $Γ= \mathbb{F}_2^k$, our general result above requires a more involved result we obtain for the natural directed analogue of Schrijver's question.

Theorems & Definitions (23)

• Conjecture 1.1: Graham, 1971
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1: Chernoff's inequality
• Lemma 3.1
• Lemma 3.2
• Definition