The smallest nonevasive graph property

Abstract

A property of n-vertex graphs called evasive if every algorithm testing this property by asking questions of the form is there an edge between vertices u and v requires, in the worst case, to ask about all pairs of vertices. Most natural graph properties are either evasive or conjectured to be such, and of the few examples of nontrivial nonevasive properties scattered in the literature the smallest one has n=6. We exhibit a nontrivial, nonevasive property of 5-vertex graphs and show that it essentially the unique such with n at most 5.