%0 Journal Article
%T Total $k$-Rainbow domination numbers in graphs
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Abdollahzadeh Ahangar, Hossein
%A Amjadi, Jafar
%A Jafari Rad, Nader
%A D. Samodivkin, Vladimir
%D 2018
%\ 06/01/2018
%V 3
%N 1
%P 37-50
%! Total $k$-Rainbow domination numbers in graphs
%K $k$-rainbow dominating function
%K $k$-rainbow domination number
%K total $k$-rainbow dominating function
%K total $k$-rainbow domination number
%R 10.22049/cco.2018.25719.1021
%X Let $kgeq 1$ 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 $vin V(G)$ with $f(v)=emptyset $, the condition $bigcup_{uin N_{G}(v)}f(u)={1,2,ldots,k}$ is fulfilled, where $N_{G}(v)$ is the open neighborhood of $v$. The weight of a $k$-RDF $f$ of $G$ is the value $omega (f)=sum _{vin V(G)}|f(v)|$. 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 ${vin V(G) mid f (v) neq emptyset}$ 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.
%U http://comb-opt.azaruniv.ac.ir/article_13683_b5784dd717acd4308580ca847ce38c2b.pdf