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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.18510 |
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| _version_ | 1866912047986376704 |
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| author | Brezovnik, Simon Poklukar, Darja Rupnik Žerovnik, Janez |
| author_facet | Brezovnik, Simon Poklukar, Darja Rupnik Žerovnik, Janez |
| contents | A $k$-rainbow dominating function ($k$RDF) of $G$ is a function that assigns subsets of $ \{1,2,...,k\}$ to the vertices of $G$ such that for vertices $v$ with $f(v)=\emptyset $ we have $\bigcup\nolimits_{u\in N(v)}f(u)=\{1,2,...,k\}$.
The weight $w(f)$ of a $k$RDF $f$ is defined as $w(f)=\sum_{v\in V(G)}\left\vert f(v)\right\vert $. The minimum weight of a $k$RDF of $G$ is called the $k$-rainbow domination number of $G$, which is denoted by $γ_{rk}(G)$. In this paper, we study the 2-rainbow domination number of the Cartesian product of two cycles. Exact values are given for a number of infinite families and we prove lower and upper bounds for all other cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18510 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The 2-rainbow domination number of Cartesian product of cycles Brezovnik, Simon Poklukar, Darja Rupnik Žerovnik, Janez Combinatorics A $k$-rainbow dominating function ($k$RDF) of $G$ is a function that assigns subsets of $ \{1,2,...,k\}$ to the vertices of $G$ such that for vertices $v$ with $f(v)=\emptyset $ we have $\bigcup\nolimits_{u\in N(v)}f(u)=\{1,2,...,k\}$. The weight $w(f)$ of a $k$RDF $f$ is defined as $w(f)=\sum_{v\in V(G)}\left\vert f(v)\right\vert $. The minimum weight of a $k$RDF of $G$ is called the $k$-rainbow domination number of $G$, which is denoted by $γ_{rk}(G)$. In this paper, we study the 2-rainbow domination number of the Cartesian product of two cycles. Exact values are given for a number of infinite families and we prove lower and upper bounds for all other cases. |
| title | The 2-rainbow domination number of Cartesian product of cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.18510 |