Feasible Graphs and Colorings

Abstract

AbstractThe problem of when a recursive graph has a recursive k‐coloring has been extensively studied by Bean, Schmerl, Kierstead, Remmel, and others. In this paper, we study the polynomial time analogue of that problem. We develop a number of negative and positive results about colorings of polynomial time graphs. For example, we show that for any recursive graph G and for any k, there is a polynomial time graph G′ whose vertex set is {0,1}* such that there is an effective degree preserving correspondence between the set of k‐colorings of G and the set of k‐colorings of G′ and hence there are many examples of k‐colorable polynomial time graphs with no recursive k‐colorings. Moreover, even though every connected 2‐colorable recursive graph is recursively 2‐colorable, there are connected 2‐colorable polynomial time graphs which have no primitive recursive 2‐coloring. We also give some sufficient conditions which will guarantee that a polynomial time graph has a polynomial time or exponential time coloring.