Recursion Theoretic Aspects of Graphs and Orders

Abstract

The sort of question to be considered has the following form. Suppose we are given arecursive graph G, with vertex set V and edge set E. By “recursive” we mean (very roughly, but good enough for almost all purposes) that there is an effective algorithm which will allow us to compute in finitely many steps whether or not a givenx is in V, and also for given x,y to determine whether or not there is an edge in E joining them. For example, if G is finite, then it is recursive (but this case really is of no interest). It is usually quite safe to assume that V is just the set ω of natural numbers, or, perhaps, a recursive subset of ω. Next suppose that G is n-colorable, by which is meant that there is a function ϕ:V→ 0,1,2,…,n-l such that if x,y ∈V are adjacent, then ϕ (x) ≠ ϕ (y). Now the question: is G recursively n-colorable? I.e., is there a recursive n-coloring ϕ ofG? If not, is there some k for which there is a recursive k-coloring ofG? More generally, what conditions can be imposed on a recursive n-colorable graph which will guarantee that it is recursively k-colorable?