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Apr 15, 2015 at 17:11 comment added Tony Huynh Yes, you are right that these results are for paths with $k$ vertices. For this question, I conveniently regarded a path of length $k$ as a path with $k$ vertices (the notion of the length of a path is somewhat ambiguous, although I usually consider the length as the number of edges).
Apr 15, 2015 at 17:08 comment added Jan Kyncl I think these results would be for paths with $k$ vertices; the question is about paths of length $k$, so $k+1$ vertices.
Apr 15, 2015 at 17:04 history answered Tony Huynh CC BY-SA 3.0