The 2-Rainbow Domination Numbers of $${oldsymbol{C}}_4Box {oldsymbol{C}}_n$$ C 4 □ C n and $${oldsymbol{C}}_8Box {oldsymbol{C}}_n$$ C 8 □ C n
National Academy Science Letters(2019)
Abstract
A k-rainbow dominating function (kRDF) of G is a function
$$f:V(G)\rightarrow {\mathcal {P}}(\{1,2,\ldots ,k\})$$
for which
$$f(v)=\emptyset $$
we have
$$\bigcup \nolimits _{u\in N(v)}f(u)=\{1,2,\ldots ,k\}$$
. The weightw(f) of a function f is defined as
$$w(f)=\sum _{v\in V(G)}\left| f(v)\right| $$
. The minimum weight of a kRDF of G is called the k-rainbow domination number of G, which is denoted by
$$\gamma _{rk}(G)$$
. In this paper, we determine the exact values of the 2-rainbow domination numbers of
$$C_4\Box C_n$$
and
$$C_8\Box C_n$$
. It follows that
$$\gamma _{r2}\not = 2\gamma $$
for graphs
$$C_4\Box C_n$$
(
$$n \ge 4$$
) and
$$C_8\Box C_n$$
(
$$n \ge 8$$
), answering in part a question raised by Brešar.
MoreTranslated text
Key words
2-rainbow domination, Domination number, Cartesian product, Cycle
AI Read Science
Must-Reading Tree
Example
![](https://originalfileserver.aminer.cn/sys/aminer/pubs/mrt_preview.jpeg)
Generate MRT to find the research sequence of this paper
Chat Paper
Summary is being generated by the instructions you defined