Weak signed Roman k-domatic number of a graph

Abstract

Let k≥ 1 be an integer. A weak signed Roman k-dominating function on a graph G isa function f:V (G)→ {-1, 1, 2} such that ΣueN[v] f(u)≥ k for everyve V(G), where N[v] is the closed neighborhood of v.A set {f1,f2, ... ,fd} of distinct weak signed Roman k-dominatingfunctions on G with the property that Σ1≤i≤d fi(v)≤ k for each ve V(G), is called a weak signed Roman k-dominating family (of functions) on G. The maximum number of functionsin a weak signed Roman k-dominating family on G is the weak signed Roman k-domatic number} of Gdenoted by dwsR k(G). In this paper we initiate the study of the weak signed Roman $k$-domatic numberin graphs, and we present sharp bounds for dwsR k(G). In addition, we determine the weak signed Roman k-domatic number of some graphs.