论文标题
罗马（k，k） - 图的数量
The Roman (k,k)-domatic number of a graph
论文作者
论文摘要
让$ k $成为一个积极的整数。图$ g $上的a {\ em roman $ k $ domination}是标签$ f：v（g）\ longrightArrow \ {0，1，2，2 \} $，使每个标签0带有标签0至少$ k $ neighen的每个顶点都有标签2的标签2。 $ k $ - $ g $上的函数，其中$ \ sum_ {i = 1}^df_i（v）\ le 2k $ in V（g）$中的每个$ v \ in v（g）$，称为a {\ em roman $（k，k）$ - primanderating family}（占主导地位）$ g $ g $ g $。罗马$（k，k）$ - $ g $上的主要功能数量是$ g $的{\ em roman $（k，k）$ - domatic number}，由$ d_ {r}^k（g）$表示。请注意，罗马$（1,1）$ - domatic Number $ d_ {r}^1（g）$是通常的roman domatic number $ d_ {r}（g）$。在本文中，我们启动了罗马$（k，k）$ - dimatic数字的研究，并以$ d_ {r}^k（g）$提出了尖锐的界限。此外，我们确定了罗马$（k，k）$ - 某些图的命令数。我们的一些结果扩展了Sheikholeslami和Volkmann在2010年对罗马Dimatic数字提供的结果。
Let $k$ be a positive integer. A {\em Roman $k$-dominating function} on a graph $G$ is a labeling $f:V (G)\longrightarrow \{0, 1, 2\}$ such that every vertex with label 0 has at least $k$ neighbors with label 2. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct Roman $k$-dominating functions on $G$ with the property that $\sum_{i=1}^df_i(v)\le 2k$ for each $v\in V(G)$, is called a {\em Roman $(k,k)$-dominating family} (of functions) on $G$. The maximum number of functions in a Roman $(k,k)$-dominating family on $G$ is the {\em Roman $(k,k)$-domatic number} of $G$, denoted by $d_{R}^k(G)$. Note that the Roman $(1,1)$-domatic number $d_{R}^1(G)$ is the usual Roman domatic number $d_{R}(G)$. In this paper we initiate the study of the Roman $(k,k)$-domatic number in graphs and we present sharp bounds for $d_{R}^k(G)$. In addition, we determine the Roman $(k,k)$-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.
