Testing graphs for colorability properties

Let P be a property of graphs. An -test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than (n 2 ) edges to make it satisfy P. The property P is called testable if for every there exists an -test for P whose total number of queries is independent of the size of the input graph. Goldreich, Goldwasser, and Ron [Property testing and its connection to learning and approximation, J ACM 45 (1998), 653–750] showed that certain graph properties, like k-colorability, admit an -test. In Alon, Fischer, Krivelevich, and Szegedy [Efficient testing of large graphs, Combinatorica 20 (2000), 451–476] a first step towards a logical characterization of the testable graph properties was made by proving that all first order properties of type “∃∀” are testable, while there exist first-order graph properties of type “∀∃” that are not testable. For proving the positive part, it was shown that all properties describable by a very general type of coloring problem are testable. While this result is tight from the standpoint of first order expressions, further steps towards the characterization of the testable graph properties can be taken by considering the coloring problem instead. It is proven here that other classes of graph properties, describable by various generalizations of the coloring notion used in Alon et al. [Combinatorica 20 (2000), 451–476], are testable, showing that this approach can broaden the understanding of the nature of the testable graph properties. The proof combines some generalizations of the methods used in Alon et al. with additional methods. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 25, 289–309, 2005