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The objective of this paper is to prove by simple construction, generalized by induction, that the bounded areas on any map, such as found on the surface of a sheet of paper or a spherical globe, can be colored completely with just 4 distinct colors. Rather than following the tradition of examining each of tens of thousands of designs that can be produced on a planar surface, the approach here is to all the ways that any given plane, or any given part of a plane, can be divided into an old portion bearing its original color as contrasted with a new portion bearing a different color and being completely separated from the former colored portion. It is shown that for every possible manner of completely carving out any piece of any planar surface by an indexical vector, the adjacent pieces of the map, defined as o nes sharing some segment of one of their borders of a length greater than 0, can always be colored with just 4 colors in a way that differentiates all the distinct pieces of the map no matter how complex or numerous the pieces may become.

Easy to express but difficult to prove, the “four-color map theorem” is a proposition plainly put by Francis Guthrie in 1852. It seemed to be an almost trivial problem for map-making, and though evidently true, it turned out to be more interesting and complex than it seemed on first look [

However, proof of the four-color theorem turned out to be more difficult than seemed likely at first blush [

Over the last several decades, it has become increasingly feasible to assess the validity of cumbersome formalized mathematical proofs by taking advantage of the speed, power, and accuracy of modern computing. However, the advantages of automated auditing of formalized proofs, such as the proofs of the four-color theorem, presents special difficulties to human auditors. Efforts along that line have led to a rephrasing of the long- standing question of “artificial intelligence”. Govindarajalulu, Bringsjord, and Taylor put that question like this: “Is it possible to apply computational power to generate entirely new proofs not previously discovered by humans?” ( [

Although the four-color theorem is believed to be true, and its complex computer- assisted proofs are believed to be valid, certain conjectures related to that theorem remain unproved and simpler proofs of the four-color theorem itself are still being sought. In 2012 Cooper, Rowland, and Zeilberger took some steps toward what they described as a “language theoretic proof” involving the parsing of binary trees. They argued that within their simple grammar the proposition “that every pair of trees has a common parse word is equivalent to the statement that every planar graph is four-co- lorable” and they also supposed that the results they achieved “are a step toward a language theoretic proof of the four color theorem” ( [

In the meantime, according to the Web of Science Core Collection, Gonthier’s 2008 computer-assisted proof of the four-color theorem has been cited at least 75 times at the time of this writing (June 14, 2016). Also, continuing interest in proofs of the four- color theorem is shown in the non-linear increase in citations of Gonthier’s proof as shown in

Therefore, in light of all the foregoing, a simple constructive proof of the four-color theorem might be of interest. In building the following proof, as in my mathematical proofs about biosemiotic entropy [

soning is mathematical reasoning, no matter how simple it may be. By diagrammatic reasoning, I mean reasoning which constructs a diagram according to a precept expressed in general terms, performs experiments upon this diagram, notes their results, assures itself that similar experiments performed upon any diagram constructed according to the same precept would have the same results, and expresses this in general terms. This was a discovery of no little importance, showing, as it does, that all knowledge without exception [including mathematical knowledge] comes from observation” ( [

Interestingly, if the proof proposed here bears up under intense critical scrutiny, it will be, I believe, because of the peculiar binary nature of the kinds of sign systems known as indexes in Peircean logic. It is their particular binary character, hinted at in the “step toward a language theoretic proof of the four color theorem” by Cooper, Rowland, and Zeilberger ( [

In the case of map-making, it is the role of every index in a well-formed map to separate any colored space contained within its scope from all other spaces. To accomplish that function, every indexical boundary in any well-formed planar map must be binary in two critical respects: for one, it forms a boundary completely separating (1) its contained space from (2) all other spaces on the map, and, for another, if during the construction of the map, the completed boundary is cut by a new index so as to form a new and distinct space on the map, the index in question can only possess at most two ends: (1) the beginning end and (2) the final end of the cut that joins its own beginning or some other edge of an existing boundary to fully enclose the new bounded space on the map. It is necessary that the two ends of the new space meet up with each other by being connected through the completed boundary enclosing the new space. If these binary aspects of every index that might be used to create a new space on a colored map are kept in mind, that peculiar binary nature will assist the reader in understanding the rigor and completeness of the following proof.

To begin, we can color any bounded surface in 1 color as in

defines a new space, a completely bounded island in the plane that is distinct from the rest of the plane. Provided the new island does not cover the whole map, in which case only 1 color would be needed, only 2 colors will be required to differentiate the emergent new island and it may take any shape whatsoever. For simplicity’s sake, however, on the map abstracted in the diagram of

Given such a 2-colored map consisting of two colored spaces in any possible arrangement, if a new island (of any shape whatsoever) should emerge completely contained within either of the existing colored spaces, as abstracted with circles in the diagram of

Given a map in any shape divided into three adjoining spaces colored distinctly in 3 colors, suppose that a new island completely contained within any of the existing colored spaces should emerge as shown by the black circles added in three places in

4^{th} color would be needed, but with just 4 colors all of the adjacent bounded parts of the whole surface could be colored and would be differentiated in all possible cases constructed in the manner described irrespective of the shapes of the adjoining spaces filling the entire map.

Yet, suppose next that a new island should emerge anywhere on the now four-co- lored map: 1) If the new island (entirely irrespective of its shape) should be completely contained within any distinctly colored area of the map, any of the other 3 contrasting colors could be chosen to differentiate that island (exactly as already shown in the abstracted diagrams in ^{th} color could be chosen for all possible cases of that complexity (as abstracted in

To see that this must be so for all possible cases, consider first the four corners problem illustrated in the actual map of the United States at the intersection of Arizona, Utah, Colorado, and New Mexico, as shown in

solution, exactly 4 colored spaces share a border with the new island colored in black. But let the new island be transformed into a different shape by cutting to its center and stretching it into a ribbon as suggested in ^{th} color to differentiate the new island. No more than 2 colored spaces can share a common border with the new island (colored in black and numbered 5) at any corner of its entire perimeter. The reason this must be so for all possible cases is bound up in the binary nature of an index.

If the end points of the cutting index the one that differentiates a new island or that divides an existing colored piece of the map into an old and a new piece should meet each other without crossing any border of any other colored space on the map, then the new island irrespective of its shape can take any of the 3 remaining colors that contrast with that of the colored space inside which the new island appears. If however, the indexical line should divide any existing colored space by intersecting any pair of existing borders, corners, or any border and corner in combination, the newly carved out out space can always be distinguished with a 4^{th} color, exactly as shown in the abstracted diagrams of

5

The constructive proof is produced by straightening the abstracted border of the new island whether it be a perfect circle as suggested abstractly in the diagram of ^{th} color rendering the new construction colorable by a maximum of 4 colors.

Further, given that it is not possible to construct any additional spaces in any portion of any planar map by any method other than the ones already examined, the proof is complete. To fault the proof it would be necessary for any would-be critic to show by any method of construction how it is possible to make an indexical cut of any bounded portion of a map in a manner that divides an existing space so as to produce a nexus of 2 adjacent corners on the perimeter of that new island that mark the intersection of more than 3 spaces with boundaries adjacent to the new island. But, that cannot be done, because we have already examined all of the possible ways of dividing any colored space on any map into an old space and a new one. We have also examined the corner problem in a manner showing that any number of corners joined at any single point on the map, or at any number of points on the perimeter of any given piece of the map, old or new, can always be resolved as already shown by the foregoing construction. Also, the simple proof presented here suggests that a “language theoretic proof” along the lines of [

By induction from the foregoing constructions, it follows that any map, including the infinitely complex sorts constructed in the complex number plane as differentiated into the fractal patterns of the Mandelbrot set, sampled in

The author declares that he has no competing interests.

Oller Jr., J.W. (2016) The Four-Color Theorem of Map-Making Proved by Construction. Open Access Lib- rary Journal, 3: e3089. http://dx.doi.org/10.4236/oalib.1103089