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 ⌈ kk′ 2(k′d−kd′) ⌉ , then every coloring of P with colors in Zk′ extends to a (k′, d′)coloring of G. If k′d − kd′ = 1 and $k′/d′% = $k/d%, then this distance threshold is nearly sharp. The proof of this includes showing that up to symmetry, there is only one (k′, d′)-coloring of the canonical (k, d)-colorable graph Gk,d in this case.