The identity holds provided that for any two vertices " A " and " B " of the graph, the number of odd Eulerian paths from " A " to " B " is the same as the number of even ones . ( Here a path is called odd or even depending on whether its edges taken in order give an odd or even permutation of the 2 " n " edges . ) Swan showed that this was the case provided the number of edges in the graph is at least 2 " n ", thus proving the Amitsur Levitzki theorem.
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