Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21283120180601Total $k$-Rainbow domination numbers in graphs37501368310.22049/cco.2018.25719.1021ENHosseinAbdollahzadeh AhangarBabol Noshirvani University of Technology0000-0002-0038-8047JafarAmjadiAzarbaijan Shahid Madani University0000-0001-9340-4773NaderJafari RadShahrood University of TechnologyVladimirD. SamodivkinUniversity of Architecture, Civil Engineering and GeodesyJournal Article20170914Let $kgeq 1$ 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. http://comb-opt.azaruniv.ac.ir/article_13683_b5784dd717acd4308580ca847ce38c2b.pdf