On the complexity of finding the chromatic number of a recursive graph I: the bounded case

Abstract

We classify functions in recursive graph theory in terms of how many queries to K (or Ø n or Ø m are required to compute them. We show that (1) binary search is optimal (in terms of the number of queries to K) for finding the chromatic number of a recursive graph and that no set of Turing degree less than O' will suffice, (2) the problem of determining if a recursive graph has a finite chromatic number is Σ 2- complete, and (3) binary search is optimal (in terms of the number of queries to Ø m ) for finding the recursive chromatic number of a recursive graph and that no set of Turing degree less than O m will suffice. We also explore how much help queries to a weaker set may provide. Some of our results have analogues in terms of asking p questions at a time, but some do not. In particular, ( p+1)- ary search is not always optimal for finding the chromatic number of a recursive graph. Most of our results are also true for highly recursive graphs, though there are some interesting differences when queries to K are allowed for free in the computation of a recursive chromatic number.