Planar Graphs: Random Walks and Bipartiteness Testing

We initiate the study of the testability of properties in\emph{arbitrary planar graphs}. We prove that \emph{bipartiteness}can be tested in constant time. The previous bound for this class of graphs was $\tilde{O}(\sqrt{n})$, and the constant-time testability was only known for planar graphs with \emph{bounded degree}. Previously used transformations of unbounded-degree sparse graphs into bounded-degree sparse graphs cannot be used to reduce the problem to the testability of bounded-degree planar graphs. Our approach extends to arbitrary minor-free graphs. Our algorithm is based on random walks. The challenge here is to analyze random walks for a class of graphs that has good separators, i.e., bad expansion. Standard techniques that use a fast convergence to a uniform distribution do not work in this case. Roughly speaking, our analysis technique self-reduces the problem of finding an odd-length cycle in a autograph $G$ induced by a collection of cycles to another multigraph $G'$ induced by a set of shorter odd-length cycles, in such a way that when a random walks finds a cycle in $G'$ with probability $p >, 0$, then it does so with probability $\lambda(p)>0$ in $G$. This reduction is applied until the cycles collapse to self-loops that can be easily detected.