On Testing Bipartiteness in the Bounded-Degree Graph Model

This memo presents a variant of the known tester of Bipartiteness in the bounded-degree graph model, which is presented in Section 9.4.1 of my book on Property Testing (hereafter referred to as the book). The purpose of this variation is to show that, when the graph is rapid mixing, Bipartiteness can be tested in O( √ k) time, rather than in Õ( √ k) time. Much of the following text is reproduced from Section 9.4.1 of the book, and the essence of the improvement is in capitalizing on half of the vertices that appear on each (2l-step long) random walk rather than using only the last vertex in each of the m walks. This is reflected in the proof of Claim 3.2, where we consider l2 · (m2 − m) collision events (rather than the ·(m2 −m) events considered in the book).