New upper bounds for the pseudoachromatic and upper irredundance numbers of a graph

Let G = (V, E) be a graph having order n = vertical bar V vertical bar vertices. The closed neighborhood of a vertex v is an element of V is the set N[v] = {u is an element of V vertical bar uv is an element of E} boolean OR {v}, and the closed neighborhood of a set S subset of V is the set N[S] = U-v is an element of S N[v]. A partition pi = {V-1, V-2,..., V-k} of the vertex set V is called a complete partition if for every 1 <= i <= j <= k, there exists a vertex u is an element of V-i that is adjacent to a vertex v is an element of V-j. The maximum order of a complete partition of a graph G is called the pseudoachromatic number psi(s)(G) of G. The upper irredundance number IR(G) equals the maximum order of a set S subset of V having the property that for every vertex v is an element of S, N[v] - N[S - {v}] not equal empty set. In this paper we discuss a variant of an old method that can be used to prove a variety of new results like the following: for any graph G of order n >= 2, IR(G) + psi(s)(G) <= n + 1.