Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21283120180601Double Roman domination and domatic numbers of graphs71771374410.22049/cco.2018.26125.1078ENLutzVolkmannRWTH Aachen University0000-0003-3496-277XJournal Article20171117A double Roman dominating function on a graph $G$ with vertex set $V(G)$ is defined in cite{bhh} as a function<br />$f:V(G)rightarrow{0,1,2,3}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least two<br />neighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must have<br />at least one neighbor $u$ with $f(u)ge 2$. The weight of a double Roman dominating function $f$ is the sum<br />$sum_{vin V(G)}f(v)$, and the minimum weight of a double Roman dominating function on $G$ is the double Roman<br />domination number $gamma_{dR}(G)$ of $G$.<br /><br />A set ${f_1,f_2,ldots,f_d}$ of distinct double Roman dominating functions on $G$ with the property that<br />$sum_{i=1}^df_i(v)le 3$ for each $vin V(G)$ is called in cite{v} a double Roman dominating family (of functions)<br />on $G$. The maximum number of functions in a double Roman dominating family on $G$ is the double Roman domatic number<br />of $G$.<br /><br />In this note we continue the study the double Roman domination and domatic numbers. In particular, we present<br />a sharp lower bound on $gamma_{dR}(G)$, and we determine the double Roman domination and domatic numbers of some<br />classes of graphs.http://comb-opt.azaruniv.ac.ir/article_13744_0b21f0f7d95e9ab99c3422cf6f3acc77.pdf