Testing Graph Properties with the Container Method

We establish nearly optimal sample complexity bounds for testing the rho-clique property in the dense graph model. Specifically, we show that it is possible to distinguish graphs on n vertices that have a rho n-clique from graphs for which at least epsilon n(2) edges must be added to form a rho n-clique by sampling and inspecting a random subgraph on only (O) over tilde(rho(3)/epsilon(2)) vertices. We also establish new sample complexity bounds for epsilon-testing k-colorability. In this case, we show that a sampled subgraph on (O) over tilde (k/epsilon) vertices suffices to distinguish k-colorable graphs from those for which any k-coloring of the vertices causes at least epsilon n(2) edges to be monochromatic. The new bounds for testing the rho-clique and k-colorability properties are both obtained via new extensions of the graph container method. This method has been an effective tool for tackling various problems in graph theory and combinatorics. Our results demonstrate that it is also a powerful tool for the analysis of property testing algorithms.