No abstract

The problem of how many colors are required for a planar map has been used as a focal point for discussions of the limits of human direct understanding vs. automated methods. It is important to continue to investigate until it is convincingly proved map coloration is an exemplary irreducible problem or until it is reduced. Meanwhile a new way of thinking about surfaces which hide N-dimensional volumes has arisen in physics employing entropy and the holographic principle. In this paper we define coloration entropy or flexibility as a count of the possible distinct colorations of a map (planar graph), and show how a guaranteed minimum coloration flexibility changes based on additions at a boundary of the map. The map is 4-colorable as long as the flexibility is positive, even though the proof method does not construct a coloration. This demonstration is successful, resulting in a compact and easily comprehended proof of the four color theorem. The use of an entropy-like method suggests comparisons and applications to issues in physics such as black holes. Therefore in conclusion some comments are offered on the relation to physics and the relation of plane-section color-ability to higher dimensional spaces. Future directions of research are suggested which may connect the concepts to not only time and distance and thus entropic gravity but also momentum.