Partitioning a graph into offensive k-alliances

An offensive k-alliance in a graph is a set S of vertices with the property that every vertex in the boundary of S has at least k more neighbors in S than it has outside of S. An offensive k-alliance S is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) offensive k-alliances. The (global) offensive k-alliance partition number of a graph Γ=(V,E), denoted by (ψkgo(Γ)) ψko(Γ), is defined to be the maximum number of sets in a partition of V such that each set is an offensive (a global offensive) k-alliance. We show that 2≤ψkgo(Γ)≤3−k if Γ is a graph without isolated vertices and k∈{2−Δ,...,0}. Moreover, we show that if Γ is partitionable into global offensive k-alliances for k≥1, then ψkgo(Γ)=2. As a consequence of the study we show that the chromatic number of Γ is at most 3 if ψ0go(Γ)=3. In addition, for k≤0, we obtain tight bounds on ψko(Γ) and ψkgo(Γ) in terms of several parameters of the graph including the order, size, minimum and maximum degree. Finally, we study the particular case of the cartesian product of graphs, showing that ψko(Γ1×Γ2)≥ψk1o(Γ1)ψk2o(Γ2), for k≤min{k1−Δ2,k2−Δ1}, where Δi denotes the maximum degree of Γi, and ψkgo(Γ1×Γ2)≥max{ψk1go(Γ1),ψk2go(Γ2)}, for k≤min{k1,k2}.