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.

• Type: Journal
• Year: 2006
• Where: JCT
• Authors: Michael O. Albertson, Douglas B. West
• Keywords: -colorable Graph, Fix Positive Integers, Graph Gk, JCT 2006