Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21287120220601Weak signed Roman k-domatic number of a graph17271416910.22049/cco.2021.26998.1178ENLutzVolkmannRWTH Aachen University0000-0003-3496-277XJournal Article20201112Let $kge 1$ be an integer. A { weak signed Roman $k$-dominating function} on a graph $G$ is<br />a function $f:V (G)longrightarrow {-1, 1, 2}$ such that $sum_{uin N[v]}f(u)ge k$ for every<br />$vin V(G)$, where $N[v]$ is the closed neighborhood of $v$.<br />A set ${f_1,f_2,ldots,f_d}$ of distinct weak signed Roman $k$-dominating<br />functions on $G$ with the property that $sum_{i=1}^df_i(v)le k$ for each $vin V(G)$, is called a<br />{ weak signed Roman $k$-dominating family} (of functions) on $G$. The maximum number of functions<br />in a weak signed Roman $k$-dominating family on $G$ is the { weak signed Roman $k$-domatic number} of $G$,<br />denoted by $d_{wsR}^k(G)$. In this paper we initiate the study of the weak signed Roman $k$-domatic number<br />in graphs, and we present sharp bounds for $d_{wsR}^k(G)$. In addition, we determine the weak signed Roman<br />$k$-domatic number of some graphs.http://comb-opt.azaruniv.ac.ir/article_14169_1e588e4245ba97d8d37a13423c97b545.pdf