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Mondrian Four-Coloring
Any planar map can be colored with four colors so that no two regions of the same color touch each other.
This Demonstration uses the following method to four-color each map:
1. A map of rectangles is converted to a cubic graph (not shown).
2. Cycle set
is a non-unique Hamiltonian cycle. Color its edges with two colors.
3. Eliminate one color of
and color the edges that are not in
in a third color. The result is a cycle set
in two colors.
4.
and
form the boundaries of two regions
and
5. Four cases leads to four possible colors for a rectangle, according to whether it is inside or outside
or
.
The method does not always work, since some cubic graphs exist that are not Hamiltonian. They are still four-colorable, but not by this method.
Contributed by:
Ed Pegg Jr
THINGS TO TRY
Automatic Animation
SNAPSHOTS
DETAILS
Ed Pegg Jr, "
Math Games: Square Packing
," Dec. 1, 2003.
RELATED LINKS
Four-Color Maps
(
Wolfram Demonstrations Project
)
Four-Color Theorem
(
Wolfram
MathWorld
)
Mrs. Perkins's Quilt
(
Wolfram
MathWorld
)
Mrs. Perkins’s Quilts
(
Wolfram Demonstrations Project
)
Remembering Martin Gardner
(
Wolfram Blog
)
Tait's Hamiltonian Graph Conjecture
(
Wolfram
MathWorld
)
PERMANENT CITATION
Ed Pegg Jr
"
Mondrian Four-Coloring
"
http://demonstrations.wolfram.com/MondrianFourColoring/
Wolfram Demonstrations Project
Published: July 21, 2010
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