You might like some of the papers about random traversals of this graph, e.g. http://front.math.ucdavis.edu/0302.5257 http://front.math.ucdavis.edu/0302.5257 an

You might like some of the papers about random traversals of this graph, e.g. http://front.math.ucdavis.edu/0302.5257 http://front.math.ucdavis.edu/0302.5257 and http://front.math.ucdavis.edu/1601.06391 http://front.math.ucdavis.edu/1601.06391