Weak multiset sequenceability and weak BHR conjecture

A subset S of a group $(G,+)$ is t-weakly sequenceable if there is an ordering (y₁, …, yₖ) of its elements such that the partial sums~s₀, s₁, …, sₖ, given by s₀ = 0 and sᵢ = ∑ⱼ₌₁ⁱ yⱼ for 1 ≤ i ≤ k, satisfy sᵢ ≠ sⱼ whenever and 1 ≤ |i-j|≤ t. In this paper, we consider the weak sequenceability problem on multisets. In particular, we are able to prove that a multiset M=[a₁λ₁,a₂λ₂,,aₙλₙ] of non-identity elements of a generic group G is t-weakly sequenceable whenever the underlying set ₁,a₂,,aₙ\ is sufficiently large (with respect to t) and the smallest prime divisor p of $|G|$ is larger than t. A related question is the one posed by the Buratti, Horak, and Rosa (briefly BHR) conjecture here considered again in the weak sense. Given a multiset M and a walk W in Cay[G: ± M], we say that W is a realization of M if Δ(W)=± M. Here we prove that a multiset M=[a₁λ₁,a₂λ₂,,aₙλₙ] of non-identity elements of G admits a realization W=(w₀,,w_) such that wᵢ≠ wⱼ whenever and 1 ≤ |i-j|≤ t assuming that |M|=λ₁+λ₂++λₙ is sufficiently large and the smallest prime divisor p of $|G|$ is larger than $t(2t+1)$.