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How to colour the US map with Yellow, Green, Red and Blue to minimize the number of states with the colour of Green

9

2

$begingroup$

I want to colour the US (only the states) map with Yellow, Green, Red and Blue. I was wondering what would be the lowest number of states with the colour of Green. We can of course use the other colours as much as we want. Please note that I want to follow the Four Color Theorem rules.

Motivation:
I am studying graph theory and I want to know if there is a way that we could limit the use of the fourth colour as much as possible. This is not a homework problem.

My attempt:
I have tried many variations and can limit it to 6 and it seems like the
minimum possible but there are infinite possibilities to try so I was wondering if there is a simpler method? Thank you in advance.

Clarification:
I am interested in only the mainland of USA. For states like Michigan that are split, I used the same colour for both parts (since they were not connected directly).

graph-theory recreational-mathematics

share|cite|improve this question

edited 1 hour ago

YuiTo Cheng

2,40641037

asked 2 hours ago

Bor KariBor Kari

3849

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  you would need to agree on a favorite version of the graph. In the actual US, there are islands, states split into disconnected regions, other things forbidden
  $endgroup$
  – Will Jagy
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  blog.computationalcomplexity.org/2006/05/… They correctly point out that three colors cannot work, as Nevada has an odd number of neighbors
  $endgroup$
  – Will Jagy
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  thank you for your suggestion, I made a few clarifications.
  $endgroup$
  – Bor Kari
  2 hours ago
  

  
  
  
  

add a comment | 

9

2

$begingroup$

I want to colour the US (only the states) map with Yellow, Green, Red and Blue. I was wondering what would be the lowest number of states with the colour of Green. We can of course use the other colours as much as we want. Please note that I want to follow the Four Color Theorem rules.

Motivation:
I am studying graph theory and I want to know if there is a way that we could limit the use of the fourth colour as much as possible. This is not a homework problem.

My attempt:
I have tried many variations and can limit it to 6 and it seems like the
minimum possible but there are infinite possibilities to try so I was wondering if there is a simpler method? Thank you in advance.

Clarification:
I am interested in only the mainland of USA. For states like Michigan that are split, I used the same colour for both parts (since they were not connected directly).

graph-theory recreational-mathematics

share|cite|improve this question

edited 1 hour ago

YuiTo Cheng

2,40641037

asked 2 hours ago

Bor KariBor Kari

3849

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  you would need to agree on a favorite version of the graph. In the actual US, there are islands, states split into disconnected regions, other things forbidden
  $endgroup$
  – Will Jagy
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  blog.computationalcomplexity.org/2006/05/… They correctly point out that three colors cannot work, as Nevada has an odd number of neighbors
  $endgroup$
  – Will Jagy
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  thank you for your suggestion, I made a few clarifications.
  $endgroup$
  – Bor Kari
  2 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

I want to colour the US (only the states) map with Yellow, Green, Red and Blue. I was wondering what would be the lowest number of states with the colour of Green. We can of course use the other colours as much as we want. Please note that I want to follow the Four Color Theorem rules.

Motivation:
I am studying graph theory and I want to know if there is a way that we could limit the use of the fourth colour as much as possible. This is not a homework problem.

My attempt:
I have tried many variations and can limit it to 6 and it seems like the
minimum possible but there are infinite possibilities to try so I was wondering if there is a simpler method? Thank you in advance.

Clarification:
I am interested in only the mainland of USA. For states like Michigan that are split, I used the same colour for both parts (since they were not connected directly).

graph-theory recreational-mathematics

share|cite|improve this question

edited 1 hour ago

YuiTo Cheng

2,40641037

asked 2 hours ago

Bor KariBor Kari

3849

$endgroup$

share|cite|improve this question

edited 1 hour ago

YuiTo Cheng

2,40641037

asked 2 hours ago

Bor KariBor Kari

3849

share|cite|improve this question

edited 1 hour ago

YuiTo Cheng

2,40641037

asked 2 hours ago

Bor KariBor Kari

3849

edited 1 hour ago

YuiTo Cheng

2,40641037

edited 1 hour ago

YuiTo Cheng

2,40641037

asked 2 hours ago

Bor KariBor Kari

3849

asked 2 hours ago

Bor KariBor Kari

3849

1 Answer
1

active

oldest

votes

6

$begingroup$

The minimum is two states that use the fourth color. Nevada and its five neighbors cannot be colored with only three colors, and similarly West Virginia and its five neighbors cannot be colored with only three colors.

But if we color Arizona and Ohio a color we use nowhere else, then the remainder of the map can be completed using only three colors:

Adjacencies between the states may be easier to see here.

share|cite|improve this answer

edited 2 hours ago

answered 2 hours ago

Misha LavrovMisha Lavrov

49.4k757108

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I need a better atlas. I'm looking at the Philadelphia area, I cannot tell what happens among Pennsylvania, New Jersey, Delaware, Maryland.
  $endgroup$
  – Will Jagy
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @WillJagy The reference I actually used to color the US was this picture of the US graph, which solves this problem.
  $endgroup$
  – Misha Lavrov
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's pretty good. A simple standard: at least one drivable road between neighbors
  $endgroup$
  – Will Jagy
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  just curious -- did you write code to do this, or did you do this by hand?
  $endgroup$
  – antkam
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @antkam By hand. I found two subgraphs where a fourth color is forced, and chose a state from each of them to color green that seemed to be a good choice. Then I just tried to color the rest with three colors - and for that, once you color the first two states, most of the rest of the map is forced, except for a few states like Maine.
  $endgroup$
  – Misha Lavrov
  35 mins ago
  
  
  

  
  
  
  

add a comment | 

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6

$begingroup$

The minimum is two states that use the fourth color. Nevada and its five neighbors cannot be colored with only three colors, and similarly West Virginia and its five neighbors cannot be colored with only three colors.

But if we color Arizona and Ohio a color we use nowhere else, then the remainder of the map can be completed using only three colors:

Adjacencies between the states may be easier to see here.

share|cite|improve this answer

edited 2 hours ago

answered 2 hours ago

Misha LavrovMisha Lavrov

49.4k757108

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I need a better atlas. I'm looking at the Philadelphia area, I cannot tell what happens among Pennsylvania, New Jersey, Delaware, Maryland.
  $endgroup$
  – Will Jagy
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @WillJagy The reference I actually used to color the US was this picture of the US graph, which solves this problem.
  $endgroup$
  – Misha Lavrov
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's pretty good. A simple standard: at least one drivable road between neighbors
  $endgroup$
  – Will Jagy
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  just curious -- did you write code to do this, or did you do this by hand?
  $endgroup$
  – antkam
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @antkam By hand. I found two subgraphs where a fourth color is forced, and chose a state from each of them to color green that seemed to be a good choice. Then I just tried to color the rest with three colors - and for that, once you color the first two states, most of the rest of the map is forced, except for a few states like Maine.
  $endgroup$
  – Misha Lavrov
  35 mins ago
  
  
  

  
  
  
  

add a comment | 

6

$begingroup$

The minimum is two states that use the fourth color. Nevada and its five neighbors cannot be colored with only three colors, and similarly West Virginia and its five neighbors cannot be colored with only three colors.

But if we color Arizona and Ohio a color we use nowhere else, then the remainder of the map can be completed using only three colors:

Adjacencies between the states may be easier to see here.

share|cite|improve this answer

edited 2 hours ago

answered 2 hours ago

Misha LavrovMisha Lavrov

49.4k757108

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I need a better atlas. I'm looking at the Philadelphia area, I cannot tell what happens among Pennsylvania, New Jersey, Delaware, Maryland.
  $endgroup$
  – Will Jagy
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @WillJagy The reference I actually used to color the US was this picture of the US graph, which solves this problem.
  $endgroup$
  – Misha Lavrov
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's pretty good. A simple standard: at least one drivable road between neighbors
  $endgroup$
  – Will Jagy
  2 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  just curious -- did you write code to do this, or did you do this by hand?
  $endgroup$
  – antkam
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  @antkam By hand. I found two subgraphs where a fourth color is forced, and chose a state from each of them to color green that seemed to be a good choice. Then I just tried to color the rest with three colors - and for that, once you color the first two states, most of the rest of the map is forced, except for a few states like Maine.
  $endgroup$
  – Misha Lavrov
  35 mins ago
  
  
  

  
  
  
  

add a comment | 

$begingroup$

The minimum is two states that use the fourth color. Nevada and its five neighbors cannot be colored with only three colors, and similarly West Virginia and its five neighbors cannot be colored with only three colors.

But if we color Arizona and Ohio a color we use nowhere else, then the remainder of the map can be completed using only three colors:

Adjacencies between the states may be easier to see here.

share|cite|improve this answer

edited 2 hours ago

answered 2 hours ago

Misha LavrovMisha Lavrov

49.4k757108

$endgroup$

share|cite|improve this answer

edited 2 hours ago

answered 2 hours ago

Misha LavrovMisha Lavrov

49.4k757108

answered 2 hours ago

Misha LavrovMisha Lavrov

49.4k757108

answered 2 hours ago

Misha LavrovMisha Lavrov

49.4k757108