Signed total 2-independence in graphs

Abstract

A function f : V(G) → {-1,1} defined on the vertices of a graph G = (V, E) is a signed total 2-independence function if the sum of its function values over any open neighborhood is at most one. That is, for every v ε V, f(N(v)) ≤ 1, where N(v) consists of every vertex adjacent to v. The weight of a signed total 2-independence function is f(V) = Σf(v), over all vertices v ε V. The signed total 2-independence number of a graph G, denoted by α_st²(G), is the maximum weight of a signed total 2-independence function of G. In this paper, we establish some upper bounds on α_st²(G) of G, and a sharp upper bound on α_st²(G) for an r-partite graph G with minimum degree δ(G) ≥ 2.