A sharp upper bound for the number of connected sets in any grid graph
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908299050352640 |
|---|---|
| author | Ma, Hongxia Jin, Xian'an Yang, Weiling Zhang, Meiqiao |
| author_facet | Ma, Hongxia Jin, Xian'an Yang, Weiling Zhang, Meiqiao |
| contents | A connected set in a graph is a subset of vertices whose induced subgraph is connected. Although counting the number of connected sets in a graph is generally a \#P-complete problem, it remains an active area of research. In 2020, Vince posed the problem of finding a formula for the number of connected sets in the $(n\times n)$-grid graph. In this paper, we establish a sharp upper bound for the number of connected sets in any grid graph by using multistep recurrence formulas, which further derives enumeration formulas for the numbers of connected sets in $(3\times n)$- and $(4\times n)$-grid graphs, thus solving a special case of the general problem posed by Vince. In the process, we also determine the number of connected sets of $K_{m}\times P_{n}$ by employing the transfer matrix method, where $K_{m}\times P_{n}$ is the Cartesian product of the complete graph of order $m$ and the path of order $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02309 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A sharp upper bound for the number of connected sets in any grid graph Ma, Hongxia Jin, Xian'an Yang, Weiling Zhang, Meiqiao Combinatorics A connected set in a graph is a subset of vertices whose induced subgraph is connected. Although counting the number of connected sets in a graph is generally a \#P-complete problem, it remains an active area of research. In 2020, Vince posed the problem of finding a formula for the number of connected sets in the $(n\times n)$-grid graph. In this paper, we establish a sharp upper bound for the number of connected sets in any grid graph by using multistep recurrence formulas, which further derives enumeration formulas for the numbers of connected sets in $(3\times n)$- and $(4\times n)$-grid graphs, thus solving a special case of the general problem posed by Vince. In the process, we also determine the number of connected sets of $K_{m}\times P_{n}$ by employing the transfer matrix method, where $K_{m}\times P_{n}$ is the Cartesian product of the complete graph of order $m$ and the path of order $n$. |
| title | A sharp upper bound for the number of connected sets in any grid graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.02309 |