In a random perfect matching of a regular bipartite graph, are all edges equally probable? Consider a d-regular bipartite graph G, for d>=1. Obviously, G contains a perfect matching. Consider a perfect matching M in G chosen uniformly at random from all perfect matchings in G. Is it the case that, for arbitrary edges e and e', the probability e is in M is the same as the probability e' is in M?

Stated differently, I am asking the following: Is it the case that, for an edge e of a regular bipartite graph G, the number of perfect matchings that include e does not depend on e?
No.

Construction: Take two copies of $K_{3,3}$, one with the nodes $\{a,b,c\} \cup \{a&#39;,b&#39;,c&#39;\}$ and the other one with the nodes $\{d,e,f\} \cup \{d&#39;,e&#39;,f&#39;\}$. Remove the edges $(c,c&#39;)$ and $(d,d&#39;)$. Add the edges $(c,d&#39;)$ and $(d,c&#39;)$.