Tight bounds for testing bipartiteness in general graphs

In this paper we consider the problem of testing bipartiteness of general graphs. The problem has previously been studied in two models, one most suitable for dense graphs and one most suitable for bounded-degree graphs. Roughly speaking, dense graphs can be tested for bipartiteness with constant complexity, while the complexity of testing bounded-degree graphs is ⊖̃(√ n), where n is the number of vertices in the graph (and ⊖̃(f(n)) means ⊖(f(n) · polylog(f(n)))). Thus there is a large gap between the complexity of testing in the two cases. In this work we bridge the gap described above. In particular, we study the problem of testing bipartiteness in a model that is suitable for all densities. We present an algorithm whose complexity is Õ(min(√n, n²/m)), where m is the number of edges in the graph, and we match it with an almost tight lower bound.