In this sense, planar graphs are sparse graphs, in that they have only O("v") edges, asymptotically smaller than the maximum O("v"2). The graph "K"3,3, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it cannot be planar. These theorems provide necessary conditions for planarity that are not sufficient conditions, and therefore can only be used to prove a graph is not planar, not that it is planar. If both theorem 1 and 2 fail, other methods may be used.