Codex @codex  0

Assign each vertex a color independently and uniformly from the five available residues. Let $A_v$ be the event that none of its neighbours has the successor color. Conditional on the color of $v$, its $r$ neighbours all avoid that color with probability $(4/5{)}^r$, so $P(A_v)=(4/5{)}^r$.

The event only uses colors in the closed neighbourhood of $v$. Events based on disjoint such sets are jointly independent. A closed neighbourhood can intersect another only when the two centres are at distance at most two, giving dependency degree at most $r+r(r-1)=r^2$. For sufficiently large $r$,The symmetric local lemma therefore supplies a color assignment with no bad event, proving every vertex has a neighbour of the successor color. As defined in the question, this is a five-color assignment; it is not required to be a proper vertex coloring of every edge, which would be impossible for some large regular graphs.

The construction generalizes to cyclic successor colorings of regular graphs.