The double Roman domatic number  of a digraph

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} $\gamma_{rk}(G)$ 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.

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Domatic Number of a Graph and its Variants (Extended Abstract)

Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity - Annals of Discrete Mathematics ◽

10.1016/s0167-5060(08)70657-5 ◽

1992 ◽

pp. 363-369

Author(s):

Bohdan Zelinka

Keyword(s):

Domatic Number

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On the Roman {2}-domatic number of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383092150052x ◽

2020 ◽

pp. 2150052

Author(s):

A. Giahtazeh ◽

H. R. Maimani ◽

A. Iranmanesh

Keyword(s):

Bipartite Graphs ◽

Domatic Number ◽

Join Of Graphs ◽

Complete Bipartite Graphs ◽

Appl Math ◽

Dominating Function

Let [Formula: see text] be a graph. A Roman[Formula: see text]-dominating function [Formula: see text] has the property that for every vertex [Formula: see text] with [Formula: see text], either [Formula: see text] is adjacent to a vertex assigned [Formula: see text] under [Formula: see text], or [Formula: see text] is adjacent to at least two vertices assigned [Formula: see text] under [Formula: see text]. A set [Formula: see text] of distinct Roman [Formula: see text]-dominating functions on [Formula: see text] with the property that [Formula: see text] for each [Formula: see text] is called a Roman[Formula: see text]-domination family (or functions) on [Formula: see text]. The maximum number of functions in a Roman [Formula: see text]-dominating family on [Formula: see text] is the Roman[Formula: see text]-domatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we answer two conjectures of Volkman [L. Volkmann, The Roman [Formula: see text]-domatic number of graphs, Discrete Appl. Math. 258 (2019) 235–241] about Roman [Formula: see text]-domatic number of graphs and we study this parameter for join of graphs and complete bipartite graphs.

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