%0 Journal Article
%T The Four-Color Theorem of Map-Making Proved by Construction
%A John W. Oller Jr.
%J Open Access Library Journal
%V 3
%N 12
%P 1-12
%@ 2333-9721
%D 2016
%I Open Access Library
%R 10.4236/oalib.1103089
%X <div style="text-align:justify;">
	<span style="line-height:1.5;">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</span><span style="line-height:1.5;">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.</span>
</div>
%K Four-Color Map Theorem
%K Proof by Construction
%K Peircean Exact (Mathematical) Logic
%U http://www.oalib.com/paper/5274653