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.
Highlights
The question of the minimum number of colors with which to color a map so that no adjacent countries sharing a border of non-zero length have the same color has enjoyed popularity since the middle 1800s when it was introduced by mathematician Augustus De Morgan, it having come to his attention through Francis and Fredrick Guthrie [1]
We consider a conceptualization in terms of state counting, very much like entropy, and explore whether state counting can be formulated as a single node induction sequence in a way that implements the obvious hiding aspects of planar maps while avoiding the back-tracking problem
Gravity should turn out to be due to a greater distribution of path-dependent state counts in the direction of mass units available for state-interactions, so that the probability of state evolution is in the direction of higher density state availability
Summary
The question of the minimum number of colors with which to color a map so that no adjacent countries sharing a border of non-zero length have the same color has enjoyed popularity since the middle 1800s when it was introduced by mathematician Augustus De Morgan, it having come to his attention through Francis and Fredrick Guthrie [1]. Pure and Applied Mathematics Journal 2018; 7(3): 37-44 whether this is intrinsic to the 4-color map problem itself, or to the ways in which it has been conceptualized. We consider a conceptualization in terms of state counting, very much like entropy, and explore whether state counting can be formulated as a single node induction sequence in a way that implements the obvious hiding aspects of planar maps while avoiding the back-tracking problem. A state is, loosely, a particular coloration of the map. Rather than claim more than an inspirational connection with entropy (though there may be), we will use the term flexibility to indicate a known number of coloration states a map may have.
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