Suppose we have a undirected graph G = (V,E) where V is the set of vertices and E is the set of edges. Suppose E consists of red colored edges and blue colored edges. Suppose we have an infinite sequence S of characters R and B.

We take a random walk starting at vertex v on G based on the sequence S as follows:

At the ith step, if S has an R at position i the walk traverses a random red edge out of the current vertex (chosen uniformly from the outgoing edges). If S has a B the walk traverses a random blue edge out of the current vertex.

We say S covers G starting at vertex v when a random walk using S starting at v covers every vertex of G.

Theorem 1 If G is a red-blue colored undirected graph which is connected in red and connected in blue and there exists an RB-sequence S such that starting at some vertex v, Pr[S covers G] < 1, then G contains a proper subgraph H such that H's vertices can be divided into two sets: U and W where there are no red edges between U and V - W and no blue edges between U and V - U.




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