Extending precolorings to circular colorings

Fix positive integers k ′ , d ′ , k, d such that k ′ / d ′ > k / d ⩾ 2 . If P is a set of vertices in a ( k , d ) -colorable graph G, and any two vertices of P are separated by distance at least 2 ⌈ k k ′ ( 2 ( k ′ d − k d ′ ) ) ⌉ , then every coloring of P with colors in Z k ′ extends to a ( k ′ , d ′ ) -coloring of G. If k ′ d − k d ′ = 1 and ⌊ k ′ / d ′ ⌋ = ⌊ k / d ⌋ , then this distance threshold is nearly sharp. The proof of this includes showing that up to symmetry, in this case there is only one ( k ′ , d ′ ) -coloring of the canonical ( k , d ) -colorable graph G k , d .