On the Finiteness of the Recursive Chromatic Number

A recursive graph is a graph whose vertex and edges sets are recursive. A highly recursive graph is a recursive graph that also has the following property: one can recursively determine the neighbors of a vertex. Both of these have been studied in the literature. We consider an intermediary notion: Let A be a set. An Arecursive graph is a recursive graph that also has the following property: one can recursively-in-A determine the neighbors of a vertex. We show that, if A is r.e. and not recursive, then there exists A-recursive graphs that are 2-colorable but not recursively k-colorable for any k. This is false for highly-recursive graphs but true for recursive graphs. Hence A-recursive graphs are closer in spirit to recursive graphs then to highly recursive graphs.

• Type: Journal
• Year: 1998
• Where: APAL
• Authors: William I. Gasarch, Andrew C. Y. Lee
• Keywords: APAL 1998, Edges Sets, Intermediary Notion, Recursive Graphs