Independent domination in the graph defined by two consecutive levels of the $n$-cube
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910497985527808 |
|---|---|
| author | Kalinowski, Thomas Leck, Uwe |
| author_facet | Kalinowski, Thomas Leck, Uwe |
| contents | Fix a positive integer $n$ and consider the bipartite graph whose vertices are the $3$-element subsets and the $2$-element subsets of $[n]=\{1,2,\dots,n\}$, and there is an edge between $A$ and $B$ if $A\subset B$. We prove that the domination number of this graph is $\binom{n}{2}-\lfloor\frac{(n+1)^2}{8}\rfloor$, we characterize the dominating sets of minimum size, and we observe that the minimum size dominating set can be chosen as an independent set. This is an exact version of an asymptotic result by Balogh, Katona, Linz and Tuza (2021). For the corresponding bipartite graph between the $(k+1)$-element subsets and the $k$-elements subsets of $[n]$ ($k\geq 3$), we provide a new construction for small independent dominating sets. This improves on a construction by Gerbner, Kezegh, Lemons, Palmer, Pálvölgyi and Patkós (2012), who studied these independent dominating sets under the name saturating flat antichains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_05327 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Independent domination in the graph defined by two consecutive levels of the $n$-cube Kalinowski, Thomas Leck, Uwe Combinatorics 05D05, 05C35 Fix a positive integer $n$ and consider the bipartite graph whose vertices are the $3$-element subsets and the $2$-element subsets of $[n]=\{1,2,\dots,n\}$, and there is an edge between $A$ and $B$ if $A\subset B$. We prove that the domination number of this graph is $\binom{n}{2}-\lfloor\frac{(n+1)^2}{8}\rfloor$, we characterize the dominating sets of minimum size, and we observe that the minimum size dominating set can be chosen as an independent set. This is an exact version of an asymptotic result by Balogh, Katona, Linz and Tuza (2021). For the corresponding bipartite graph between the $(k+1)$-element subsets and the $k$-elements subsets of $[n]$ ($k\geq 3$), we provide a new construction for small independent dominating sets. This improves on a construction by Gerbner, Kezegh, Lemons, Palmer, Pálvölgyi and Patkós (2012), who studied these independent dominating sets under the name saturating flat antichains. |
| title | Independent domination in the graph defined by two consecutive levels of the $n$-cube |
| topic | Combinatorics 05D05, 05C35 |
| url | https://arxiv.org/abs/2311.05327 |