On the Finiteness of the Recursive Chromatic Number

Abstract A recursive graph is a graph whose vertex and edge 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 A-recursive 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 than to highly recursive graphs.