Understanding the Four Color Theorem

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Discussion Overview

The discussion revolves around the Four Color Theorem, specifically focusing on the definition and implications of "contiguous regions" in the context of the theorem. Participants explore the meaning of contiguity, the nature of regions, and the conditions under which the theorem applies.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants seek clarification on what constitutes a "contiguous region," questioning whether regions need to be in contact and how boundaries affect this definition.
  • One participant explains that contiguous means each region must be connected as a single piece, without gaps, and emphasizes that non-contiguous regions would require lifting a paintbrush to color.
  • Another participant raises a hypothetical scenario involving a circle with multiple regions connected to it, seeking to understand if this configuration would be considered contiguous.
  • There is a discussion about the implications of regions touching at a single point, with a participant noting that such arrangements do not meet the criteria for contiguity.
  • Some participants express surprise at the lack of limitations on the shapes that can be drawn, suggesting that as long as there are no gaps, various configurations are permissible.

Areas of Agreement / Disagreement

Participants generally agree on the definition of contiguous regions but express differing interpretations of specific scenarios and conditions under which the theorem applies. The discussion remains unresolved regarding the nuances of certain configurations.

Contextual Notes

Participants mention potential limitations regarding the definition of regions and the implications of non-contiguous arrangements, but these aspects remain open for further exploration.

cragar
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I was reading about the four color theorem and I am not sure I understand the statement of the theorem. On wiki it says that you can divide the plane into contiguous regions. I am not sure what they mean by contiguous region. Does that mean that the shapes need to be in contact with one another. Does it matter how many shapes another shape has at its boundaries.
 
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Contiguous means each region is connected (only one piece). Also as far as the four color theorem is concerned, there is no vacancy. There are no other limits.
 
Im still not sure what you mean when you say that each region is connected (only one piece ) Like what if I had a circle in the middle and then like 4 regions connected to it, would that work?
 
cragar said:
Im still not sure what you mean when you say that each region is connected (only one piece ) Like what if I had a circle in the middle and then like 4 regions connected to it, would that work?

only if the circle in the middle is to be colored-in, too. "gaps" aren't fair.

the reason contiguous regions are specified is to avoid the situation that crops up with the continental United States and Alaska, the region "the United States of America" is not contiguous (it has non-touching "pieces").
 
Suppose you want to paint a region a particular color. If it is a contiguous region, you could paint the whole region without lifting your paint brush off the surface and putting it down again. It if is a non-contiguous region, you would be forced to pick up the paintbrush and put it down somewhere else.

Another way to explain it is that "contiguous" means you can't "cheat" by saying "I'm going to say these separate regions are all part of one big region, so you have to paint them all the same color". If you could make up "rules" like that, you could invent maps where the minimum number of colors required was arbitrarily large, because you could draw a map where every "region" shared a boundary with every other "region".
 
so basically i could take a sheet of paper . And draw any crazy shapes on it and as many as i want, just as long as there are no gaps on the page.
 
Also the situation where regions touch in a single point- as a circle divided into many "pies"- is not valid.
 
cragar said:
so basically i could take a sheet of paper . And draw any crazy shapes on it and as many as i want, just as long as there are no gaps on the page.
That's right. And I'll bet you can't draw any combination of 'crazy shapes' that take more than four colors.
 
I thought there were more limitations on the map, but it seems like there aren't that many.
Thats a pretty interesting theorem.
 

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