Avoidance couplings on non‐complete graphs

A coupling of random walkers on the same finite graph, who take turns sequentially, is said to be an avoidance if never collide. Previous studies these processes have focused almost exclusively complete graphs, in particular how many can include. For other apart from special cases, it has been unsettled whether even two noncolliding simple coupled. In this article, we construct such a (i) any d-regular graph avoiding fixed subgraph depending d; and (ii) square-free with minimum degree at least three. corollary first result that uniformly regular n vertices admits high probability.