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.

For a finite graph of maximum degree $\Delta \ge 1$, a random independent edge coloring and the Lovász local lemma prove existence of a coloring with no repetitively colored four-edge path when $k\ge 8{\Delta }^{3/2}$, and no repetitively colored six-edge path when $k\ge 5{\Delta }^{5/3}$.

A fixed edge lies in at most $\ell {\Delta }^{\ell -1}$ unoriented paths of length $\ell$. Hence a four-edge event has a closed dependency neighbourhood of size at most $16{\Delta }^3$ and probability $k^{-2}$; the symmetric local-lemma product is at most $16e{\Delta }^3/k^2$. The corresponding six-edge quantities are $36{\Delta }^5$, $k^{-3}$ and $36e{\Delta }^5/k^3$. The stated constants make both products less than one.

Both lengths can be excluded simultaneously with $k\ge C{\Delta }^{5/3}$ for an absolute constant $C$. Apply the asymmetric Lovász local lemma with parameters $x_4=2/k^2$, $x_6=2/k^3$. A path of length $\ell \in \{4,6\}$ meets at most $4\ell {\Delta }^3$ four-edge paths and $6\ell {\Delta }^5$ six-edge paths. The sums of neighbour parameters are therefore bounded by constants times $C^{-2}{\Delta }^{-1/3}+C^{-3}$. Choosing $C$ sufficiently large makes both products in the local-lemma inequalities at least one half, establishing the claim.