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 = (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. In this model a testing algorithm is allowed to ask neighbor queries (i.e., "what is the i-th neighbor of vertex v"), vertex-pair queries (i.e., "is there an edge between vertices v and u"), and degree queries (i.e., "what is the degree of vertex v"). Our ma...

• Type: Journal
• Year: 2008
• Where: SIAMDM
• Authors: Noga Alon, Tali Kaufman, Michael Krivelevich, Dana Ron
• Keywords: Fixed Subgraph H, Lower Bounds, SIAMDM 2008, Subgraph H