 
Summary: Extending Precolorings to Circular Colorings
Michael O. Albertson
and Douglas B. West
September, 2005
Abstract
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 dkd ) , 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.
1 Introduction
A precoloring extension problem asks whether a coloring of a specified vertex subset P in a
graph G extends to a "good" coloring of all of G. Such problems are usually NPcomplete
[11]. One might hope that if the vertices of P were far enough apart, then there would be
enough flexibility to extend a given precoloring to an optimal coloring of G (that is, a proper
coloring with the minimum possible number (G) of colors).
More generally, let G[P] denote the subgraph of G induced by a vertex set P, and define
d(P) to be the minimum distance between components of G[P]. A precoloring extension
