Natural SciencesMathematicsCombinatoricsDiscrete MathematicsTheory Of ComputationEnumerative CombinatoricsGraph ColoringGraph TheoryRecursion TheoryChromatic PolynomialsColoring Planar GraphsColoring TheoremsPlanar Graph AlgorithmsRecursive Functions
Abstract We are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k , has no recursive k -coloring; every decidable graph of genus p ≥ 0 has a recursive 2 (x(p) − 1)-coloring, where x(p) is the least number of colors which will suffice to color any graph of genus p ; for every k ≥ 3 there is a k -colorable, decidable graph with no recursive k -coloring, and if k = 3 or if k = 4 and the 4-color conjecture fails the graph is planar; there are degree preserving correspondences between k -colorings of graphs and paths through special types of trees which yield information about the degrees of unsolvability of k -colorings of graphs.