  Assume that there is a free group action of automorphisms on a bipartite
graph. If there is a perfect matching on the factor graph, then obviously there
is a perfect matching on the graph. Surprisingly, the reversed is also true for
amenable groups: if there is a perfect matching on the graph, there is also a
perfect matching on the factor graph, i. e. a group invariant ("symmetric")
perfect matching on the graph.
