A topological approach to evasiveness

The complexity of a digraph property is the number of entries of the vertex adjacency matrix of a digraph which must be examined in worst case to determine whether the digraph has the property. Rivest and Vuillemin proved the result (conjectured by Aanderaa and Rosenberg) that every graph property that is monotone (preserved by addition of edges) and nontrivial (holds for some but not all graphs) has complexity θ(v2) where v is the number of vertices. Karp conjectured that every such property is evasive, i.e., requires that every entry of the incidence matrix be examined. In this paper it is shown that Karp´s conjecture follows from another conjecture concerning group actions on topological spaces. A special case of this conjecture is proved and applied to prove Karp´s conjecture for the case of properties of graph and digraph properties on a prime power number of vertices.