# On the rainbow domination number of graphs

###
Polona Repolusk

University of Maribor, Faculty of Natural Sciences and Mathematics

####
Janez Žerovnik

University of Ljubljana, Faculty of Mechanical Engineering

####
Zehui Shao

Chengdu University, School of Information Science and Technology

####
Meilian Liang

Guangxi University, School of Mathematics and Information Science

####
Chuang Yin

Guangxi Academy of Sciences

####
Xiaodong Xu

Guangxi Academy of Sciences

PDF

**Minisymposium:**
GRAPH DOMINATION

**Content:**
Given a graph $G$ and a set of $t$ colors $\lbrace 1,2,\ldots , t \rbrace$, assume that we assign an arbitrary subset of these colors to each vertex of $G$. If we require that each vertex to which an empty set is assigned has in its neighborhood all $t$ colors, then this assignment is called a $t$-rainbow dominating function of the graph $G$. The corresponding invariant $\gamma_{rt}(G)$, which is the minimum sum of numbers of assigned colors over all vertices of $G$, is called the $t$-rainbow domination number of $G$. In this talk, bounds for the $t$-rainbow domination number of an arbitrary graph will be presented. For a special case of the 3-rainbow domination numbers, bounds or exact values will be presented for several classes of graphs including paths, cycles and the generalized Petersen graphs $P(n,k)$.