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Donald Knuth, in a draft of a coming volume of The Art of ComputerProgramming, has recently conjectured that in Depth-First Search of a randomdigraph with geometric outdegree distribution, the numbers of back and forwardarcs have the same distribution.  We show that this conjecture is equivalent to an equality between twogenerating functions defined by different recursions.  Unfortunately, we have not been able so use this to prove the conjecture,which still is open, but we hope that this note will inspire others to succeedwith the conjecture.

On Knuth's conjecture for back and forward arcs in Depth First Search in a random digraph with geometric outdegree distribution