
Graph Coloring, or Proof by Crayon - ColinWright
http://jeremykun.wordpress.com/2011/07/14/graph-coloring-or-proof-by-crayon/
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Ogre
My dad was a mathemetician ("was" as in he died in 2001, otherwise I'd be
sending this to him to comment on). I never understood most of what he did,
but I know that some of the work he did was based on or related to the four-
color proof referenced here. I once had a slightly less math intensive book
(but still very academic) mainly about the four color theorem that referenced
one of his papers as well, but alas I don't have it handy to find the relevant
quote. I don't know where my copy went, but I bet my mom still has one.

I don't know that I have a point to this really other than to say that this
post reminded me of my Dad, and all the things I haven't been able to send him
links to for the last 11 years.

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AbsoluteZeroFF
I find it amazing that people with such a deep and detailed grasp of
mathematics such as this person are able to also express very advanced
concepts in layman's terms through real-world examples and practical
applications.

I'm only a semi-advanced secondary school student, so many of the concepts are
way over my head at this point. Things like this article serve as catalysts to
curiosity and are inspirations to pursue the study of the mathematics.

~~~
richforrester
That's pretty much what I came to post here, but way more eloquent than when I
would've written it :)

Articles like this make me realize that what was the most boring subject in
school 10 years ago, is now quickly becoming my favorite tool in the world.
Gotta love mathematics.

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pjungwir
If people want more like this, _Introductory Graph Theory_ by Chartrand is an
enjoying and easy read. In fact it looks like Dover has all kinds of short
treatments of math subjects suitable to the layperson. And if you don't know
about Martin Gardner, you really must check him out! _The Colossal Book of
Mathematics_ and _The Night is Large_ are both wonderful big collections, and
there are tons of smaller sets of essays.

~~~
jnotarstefano
Another enjoyable book on this topic is "The Mathematical Coloring Book" by
Alexander Soifer: [http://www.amazon.com/Mathematical-Coloring-Book-
Mathematics...](http://www.amazon.com/Mathematical-Coloring-Book-Mathematics-
Colorful/dp/0387746404)

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cperciva
In case anyone is wondering: Paris is not 3-colourable. The proof is trivial,
by looking at region 6 and its neighbours.

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agrona
Specifically:

6 - R

7 - G

1 - B

15 - B

16 - R

8 - Cannot be R G or B

~~~
mgallivan
Is there any way to translate between this rather visual proof and the degrees
of 8's neighbouring vertices?

I tried to find if there was some theory behind it but all I could really find
was "saturation degree".

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ColinWright
Not really, no. There are many, many results about when something is and is
not three-colorable, but in the end, graph three coloring is NP-Complete.

~~~
mgallivan
Thanks!

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dmerfield
_So indeed it is true! We may color every map with just four colors!_

Not true. A map of a country with four or more enclaves (e.g. late 19th
century China) would not meet this condition.

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cookingrobot
Why can't you color every enclave the same color - something different from
the province they're inside of?

The traditional 4-color map rule only says that each neighboring region has a
different color, not that the colors are meaningful.

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ColinWright
In mapping, enclaves are usually colored the same as the "parent" country to
which they belong. It's an additional restriction that's actually well catered
for in the generic graph coloring context, as opposed to the "map" coloring of
the original problem (which did not bother with the enclave issue).

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jscipione
I don't understand why the Costa Rica example is 3-colorable. The green
province in the middle touches 5 other provinces so it seems like it should be
5-colorable.

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ColinWright
Consider a pizza sliced into six pieces. Each piece touches only their
neighbor on each side, so you can color them alternately red and blue. Now
suppose someone carved an island in the middle around an olive that no one
wants. That touches all six, but can still be colored green, so we use only
three colors in total, despite the middle "region" having six neighbors.

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Evbn
I don't know Paris specifically, but but nautilusy spiral happens (in
anthropology, and also biology) when new growth happens that the position
closest to the center, which is a natural goal when proximity is a desirable
feature (minimized construction costs, maximize access to community, minimize
outer perimeter for defense, etc)

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esrauch
I'm not sure this case can actually be said to be nautilusy rather than simply
uneven concentric circles.

