@article {
author = {Abdollahzadeh Ahangar, Hossein and Amjadi, Jafar and Jafari Rad, Nader and D. Samodivkin, Vladimir},
title = {Total $k$-Rainbow domination numbers in graphs},
journal = {Communications in Combinatorics and Optimization},
volume = {3},
number = {1},
pages = {37-50},
year = {2018},
publisher = {Azarbaijan Shahid Madani University},
issn = {2538-2128},
eissn = {2538-2136},
doi = {10.22049/cco.2018.25719.1021},
abstract = {Let $k\geq 1$ be an integer, and let $G$ be a graph. A $k$-rainbow dominating function (or a {$k$-RDF) of $G$ is a function $f$ from the vertex set $V(G)$ to the family of all subsets of $\{1,2,\ldots ,k\}$ such that for every $v\in V(G)$ with $f(v)=\emptyset $, the condition $\bigcup_{u\in N_{G}(v)}f(u)=\{1,2,\ldots,k\}$ is fulfilled, where $N_{G}(v)$ is the open neighborhood of $v$. The weight of a $k$-RDF $f$ of $G$ is the value $\omega (f)=\sum _{v\in V(G)}|f(v)|$. A $k$-rainbow dominating function $f$ in a graph with no isolated vertex is called a total $k$-rainbow dominating function if the subgraph of $G$ induced by the set $\{v\in V(G) \mid f (v) \neq \emptyset\}$ has no isolated vertices. The total $k$-rainbow domination number of $G$, denoted by $\gamma_{trk}(G)$, is the minimum weight of a total $k$-rainbow dominating function on $G$. The total $1$-rainbow domination is the same as the total domination. In this paper we initiate the study of total $k$-rainbow domination number and we investigate its basic properties. In particular, we present some sharp bounds on the total $k$-rainbow domination number and we determine the total $k$-rainbow domination number of some classes of graphs. },
keywords = {$k$-rainbow dominating function,$k$-rainbow domination number,total $k$-rainbow dominating function,total $k$-rainbow domination number},
url = {http://comb-opt.azaruniv.ac.ir/article_13683.html},
eprint = {http://comb-opt.azaruniv.ac.ir/article_13683_b5784dd717acd4308580ca847ce38c2b.pdf}
}