Signed total k-independence in graphs

Abstract

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 total fc-independence function if the sum of its function values over any neighborhood is at most fc - 1. That is, σxϵN(v) f(x) ≤ k - 1 for every v ϵ V(G), where N(v) consists of every vertex adjacent to v. The weight of a signed total k-independence function f is ω(f) = σvϵV(G)f(v) The maximum of weights w(f), taken over all signed total k-independence functions f on G, is the signed total k-independence number ak st(G) of G. In this work, we mainly present upper bounds on aks(t)(G), as for example aft(G) < n-2[(δ(G)+2-k)/2J1and we prove the Nordhaus- Gaddum type inequality aJt(G) + akst(G) + 2k- 3, where n is the order, δ(G) the maximum degree and G the complement of the graph G. Some of our results imply known bounds on the signed total 2-independence number. © 2015 Utilitas Mathematics.