论文标题
在$ k $ - 中间图中的rainbow统治
On $k$-rainbow domination in middle graphs
论文作者
论文摘要
让$ g $为有限的简单图，带有顶点套装$ v（g）$和edge set $ e（g）$。函数$ f：v（g）\ rightarrow \ nathcal {p}（\ {1，2，2，\ dotsc，k \}）$是a \ textit {$ k $ -rainbow统治函数$ g $，如果每个顶点$ v \ in V（g）$ f（g）$ f（v）$ f（v） n（v）} f（u）= \ {1，2，\ dotsc，k \} $。 $ k $ -rainbow主导函数的重量是$ \ sum_ {v \ in v（g）} | f（v）| $。 \ textit {$ k $ -rainbow统治号码} $γ_{rk}（g）$是$ g $上$ k $ -rainbow统治功能的最小重量。在本文中，我们在中间图中启动了$ k $ rainbow统治数字的研究。我们定义了中间$ k $ rainbow主导函数的概念，获得与之相关的一些界限，并确定某些图形类别的中间$ 3 $ rainbow统治数。我们还为中间和下界提供了$ 3 $ 3 $ rainbow统治的树木数量。此外，我们确定了路径和周期中间图的$ 3 $ -RAINBOW DIMATIC号码。
Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $γ_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.
