Using martingale techniques and comparison with the generalized Urn scheme, it is shown that the edge reinforced random walk on a graph of bounded degree, with the weight function $W(k) = k^\rho,\,\rho > 1 $, traverses (crosses) a random attracting edge at all large times. If the graph is a triangle, the above result is in agreement with a conjecture of Sellke.
"Attracting edge property for a class of reinforced random walks." Ann. Probab. 31 (3) 1615 - 1654, July 2003. https://doi.org/10.1214/aop/1055425792