Testing triangle-freeness in general graphs

In this paper we consider the problem of testing whether a graph is triangle-free, and more generally, whether it is H-free, for a fixed subgraph H. The algorithm should accept graphs that are triangle-free and reject graphs that are far from being triangle-free in the sense that a constant fraction of the edges should be removed in order to obtain a triangle-free graph. The algorithm is allowed a small probability of error. This problem has been studied quite extensively in the past, but the focus was on dense graphs, that is, when d = Θ(n), where d is the average degree in the graph and n is the number of vertices. Here we study the complexity of the problem in general graphs, that is, for varying d. Our main finding is a lower bound of Ω(n ^1/3) on the necessary number of queries that holds for every d < n^1-v(n), where v(n) = 0(1). Since when d = Θ(n) the number of queries sufficient for testing has been known to be independent of n, we observe an abrupt, thresholdlike behavior of the complexity of testing around n. This lower bound holds for testing H-freeness of every non-bipartite subgraph H. Additionally we provide sub-linear upper bounds for testing triangle-freeness that are at most quadratic in the stated lower bounds, and we describe a transformation from certain one-sided error lower bounds for testing subgraph-freeness to two-sided error lower bounds. Finally, in the course of our analysis we show that dense random Cayley graphs behave like quasi-random graphs in the sense that relatively large subsets of vertices have the "correct" edge density. The result for subsets of this size cannot be obtained from the known spectral techniques that only supply such estimates for much larger subsets.