 Repetitively colored path

An even-length graph path in an edge coloring is repetitively colored when its first half and second half have the same color sequence in the same order. For a path with distinct edges colored independently and uniformly from $k$ colors, the probability of a repetition on $2r$ edges is $k^{-r}$: after the first $r$ colors, the last $r$ colors are prescribed. Reversing the path preserves this property, so the corresponding bad events may be indexed by unoriented paths.