Signed Total k-independence in Digraphs

Let k >= 2 be an integer. A function f : V(D) -> {-1, 1} defined on the vertex set V(D) of a digraph D is a signed total k-independence function if Sigma(x is an element of N-(v)) f(x) <= k - 1 for each v is an element of V(D), where N-(v) consists of all vertices of D from which arcs go into v. The weight of a signed total k-independence function f is defined by w(f) = Sigma(x is an element of V(D)) f(x). The maximum of weights w(f), taken over all signed total k-independence functions f on D, is the signed total k-independence number alpha(k)(st) (D) of D. In this work, we mainly present upper bounds on alpha(k)(st) (D), as for example alpha(k)(st) (D) <= n - 2 [(Lambda(-) + 1 - k)/2] and alpha(k)(st) (D) <= Lambda(+) + 2k - delta(+) - 2/Delta(+) + delta(+). n, where n is the order, Delta(-) the maximum indegree and Delta(+) and delta(+) are the maximum and minimum outdegree of the digraph D. Some of our results imply well-known properties on the signed total 2-independence number of graphs.