Let (W,S) be a Coxeter system, T={wsw−1|s∈S, w∈W} its set of reflections, ≺ any total reflection order, and Γ the undirected Bruhat graph. We consider the natural labeling of the edges of Γ which assigns to the edge {v,w} the reflection vw−1. A path on Γ, i.e., a sequence v1,…,vk such that viv−1i+1∈T for i=1,…,k−1, is called T-increasing if v1v−12≺···≺vk−1v−1k. T-increasing paths play an important role in the computation of both the Kazhdan–Lusztig and the R-polynomials of W. We prove that if W is finite or is an affine Weyl group, then any T-increasing path is self-avoiding, i.e., it has no self-intersection points.