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 , 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, in this case there is only one (k′,d′)-coloring of the canonical (k,d)-colorable graph Gk,d.