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 = ₉=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=a₁^₁, a₂^₂, , aₙ^ₙ of non-identity elements of a generic group G is t-weakly sequenceable whenever the underlying set \a₁, 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 CayG: M, we say that W is a realization of M if (W) = M. Here we prove that a multiset M=a₁^₁, a₂^₂, , aₙ^ₙ 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|=₁+₂++ₙ is sufficiently large and the smallest prime divisor p of |G| is larger than t (2t+1).