Weak multiset sequenceability and weak BHR conjecture

Abstract

A subset $S$ of a group $(G,+)$ is $t$-weakly sequenceable if there is an ordering $(y_1,\dots ,y_k)$ of its elements such that the partial sums~$s_0,s_1,\dots ,s_k$, given by $s_0=0$ and $s_i={\sum }_{j=1}^i{y}_j$ for $1\le i\le k$, satisfy $s_i\ne s_j$ whenever and $1\le |i-j|\le t$. 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 $w_i\ne w_j$ whenever and $1\le |i-j|\le t$ assuming that $|M|={\lambda }_1+{\lambda }_2+\cdots +{\lambda }_n$ is sufficiently large and the smallest prime divisor $p$ of $|G|$ is larger than $t(2t+1)$.

arXiv admin note: text overlap with arXiv:2306.02721