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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2403.06781 |
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| _version_ | 1866917610558324736 |
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| author | Costa, Simone |
| author_facet | Costa, Simone |
| contents | 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 $M=[a_1^{λ_1},a_2^{λ_2},\dots,a_n^{λ_n}]$ 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 $Δ(W)=\pm M$. Here we prove that a multiset $M=[a_1^{λ_1},a_2^{λ_2},\dots,a_n^{λ_n}]$ 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 $|M|=λ_1+λ_2+\dots+λ_n$ is sufficiently large and the smallest prime divisor $p$ of $|G|$ is larger than $t(2t+1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06781 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak multiset sequenceability and weak BHR conjecture Costa, Simone Combinatorics 05C25, 05C38, 05D40 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 $M=[a_1^{λ_1},a_2^{λ_2},\dots,a_n^{λ_n}]$ 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 $Δ(W)=\pm M$. Here we prove that a multiset $M=[a_1^{λ_1},a_2^{λ_2},\dots,a_n^{λ_n}]$ 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 $|M|=λ_1+λ_2+\dots+λ_n$ is sufficiently large and the smallest prime divisor $p$ of $|G|$ is larger than $t(2t+1)$. |
| title | Weak multiset sequenceability and weak BHR conjecture |
| topic | Combinatorics 05C25, 05C38, 05D40 |
| url | https://arxiv.org/abs/2403.06781 |