We study an old question in combinatorial group theory which can be traced back to a conjecture of Graham from 1971. Given a group, and some subset S, is it possible to permute S as s₁, s₂, , sd so that the partial products ₁ ₈ ₓ sᵢ, t d are all distinct? Most of the progress towards this problem has been in the case when is a cyclic group. We show that for any group and any S, 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 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 = F₂ᵏ, our general result above requires a more involved result we obtain for the natural directed analogue of Schrijver's question.
Bucić et al. (Mon,) studied this question.