Signed k-independence in graphs

Let k ≥ 2 be an integer. A function f: V(G) → {−1, 1} defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k − 1. That is, Σx∈N[v]f(x) ≤ k − 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σv∈V(G)f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence number αsk(G) of G.In this work, we mainly present upper bounds on αsk (G), as for example αsk(G) ≤ n − 2⌈(Δ(G) + 2 − k)/2⌉, and we prove the Nordhaus-Gaddum type inequality $$\alpha _S^k \left( G \right) + \alpha _S^k \left( {\bar G} \right) \leqslant n + 2k - 3$$, where n is the order, Δ(G) the maximum degree and $$\bar G$$ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.