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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2503.05442 |
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| _version_ | 1866908412206383104 |
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| author | Luo, Yi-Lu Deng, Yun-Ping Sun, Yuan |
| author_facet | Luo, Yi-Lu Deng, Yun-Ping Sun, Yuan |
| contents | Let $G$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. Let $T$ be a subset of $ V(G)$ with cardinality $|T|\geq2$. A path connecting all vertices of $T$ is called a $T$-path of $G$. Two $T$-paths $P_i$ and $P_j$ are said to be internally disjoint if $V(P_i)\cap V(P_j)=T$ and $E(P_i)\cap E(P_j)=\emptyset$. Denote by $π_G(T)$ the maximum number of internally disjoint $T$- paths in G. Then for an integer $\ell$ with $\ell\geq2$, the $\ell$-path-connectivity $π_\ell(G)$ of $G$ is formulated as $\min\{π_G(T)\,|\,T\subseteq V(G)$ and $|T|=\ell\}$. In this paper, we study the $3$-path-connectivity of $n$-dimensional bubble-sort star graph $BS_n$. By deeply analyzing the structure of $BS_n$, we show that $π_3(BS_n)=\lfloor\frac{3n}2\rfloor-3$, for any $n\geq3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05442 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | 3-path-connectivity of bubble-sort star graphs Luo, Yi-Lu Deng, Yun-Ping Sun, Yuan Combinatorics Let $G$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. Let $T$ be a subset of $ V(G)$ with cardinality $|T|\geq2$. A path connecting all vertices of $T$ is called a $T$-path of $G$. Two $T$-paths $P_i$ and $P_j$ are said to be internally disjoint if $V(P_i)\cap V(P_j)=T$ and $E(P_i)\cap E(P_j)=\emptyset$. Denote by $π_G(T)$ the maximum number of internally disjoint $T$- paths in G. Then for an integer $\ell$ with $\ell\geq2$, the $\ell$-path-connectivity $π_\ell(G)$ of $G$ is formulated as $\min\{π_G(T)\,|\,T\subseteq V(G)$ and $|T|=\ell\}$. In this paper, we study the $3$-path-connectivity of $n$-dimensional bubble-sort star graph $BS_n$. By deeply analyzing the structure of $BS_n$, we show that $π_3(BS_n)=\lfloor\frac{3n}2\rfloor-3$, for any $n\geq3$. |
| title | 3-path-connectivity of bubble-sort star graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.05442 |