Weak multiset sequenceability and weak BHR conjecture

Abstract

A subset $S$ of a group $(G,+)$ is $t$-weakly sequenceable if there is an ordering $(y_1, \ldots, y_k)$ of its elements such that the partial sums~$s_0, s_1, \ldots, s_k$, given by $s_0 = 0$ and $s_i = \sum_{j=1}^i y_j$ for $1 \leq i \leq k$, satisfy $s_i \neq s_j$ whenever and $1 \leq |i-j|\leq t$. In this paper, we consider the weak sequenceability problem on multisets. In particular, we are able to prove that a multiset $M=[a_1^{\lambda_1},a_2^{\lambda_2},\dots,a_n^{\lambda_n}]$ of non-identity elements of a generic group $G$ is $t$-weakly sequenceable whenever the underlying set $\{a_1,a_2,\dots,a_n\}$ 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 $Cay[G: \pm M]$, we say that $W$ is a realization of $M$ if $\Delta(W)=\pm M$. Here we prove that a multiset $M=[a_1^{\lambda_1},a_2^{\lambda_2},\dots,a_n^{\lambda_n}]$ of non-identity elements of $G$ admits a realization $W=(w_0,\dots,w_{\ell})$ such that $w_i\neq w_j$ whenever and $1 \leq |i-j|\leq t$ assuming that $|M|=\lambda_1+\lambda_2+\dots+\lambda_n$ is sufficiently large and the smallest prime divisor $p$ of $|G|$ is larger than $t(2t+1)$.