Four Colour Theorem
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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
asked 1 hour ago
Bor KariBor Kari
3749
$endgroup$
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
asked 1 hour ago
Bor KariBor Kari
3749
$endgroup$
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
1 hour 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
1 hour ago
$begingroup$
thank you for your suggestion, I made a few clarifications.
$endgroup$
– Bor Kari
55 mins 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
asked 1 hour ago
Bor KariBor Kari
3749
$endgroup$
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
graph-theory recreational-mathematics
asked 1 hour ago
Bor KariBor Kari
3749
asked 1 hour ago
Bor KariBor Kari
3749
edited 1 hour ago
Bor Kari
asked 1 hour ago
Bor KariBor Kari
3749
asked 1 hour ago
Bor KariBor Kari
3749
asked 1 hour ago
Bor KariBor Kari
3749
3749
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
1 hour 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
1 hour ago
$begingroup$
thank you for your suggestion, I made a few clarifications.
$endgroup$
– Bor Kari
55 mins ago
add a comment |
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
1 hour 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
1 hour ago
$begingroup$
thank you for your suggestion, I made a few clarifications.
$endgroup$
– Bor Kari
55 mins ago
1
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
1 hour ago
$begingroup$
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
1 hour ago
1
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
1 hour ago
$begingroup$
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
1 hour ago
$begingroup$
thank you for your suggestion, I made a few clarifications.
$endgroup$
– Bor Kari
55 mins ago
$begingroup$
thank you for your suggestion, I made a few clarifications.
$endgroup$
– Bor Kari
55 mins ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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.
answered 1 hour ago
Misha LavrovMisha Lavrov
49.3k757108
$endgroup$
$begingroup$
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
35 mins 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
34 mins ago
$begingroup$
That's pretty good. A simple standard: at least one drivable road between neighbors
$endgroup$
– Will Jagy
29 mins ago
$begingroup$
just curious -- did you write code to do this, or did you do this by hand?
$endgroup$
– antkam
6 secs ago
add a comment |
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1 Answer
1
active
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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.
answered 1 hour ago
Misha LavrovMisha Lavrov
49.3k757108
$endgroup$
$begingroup$
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
35 mins 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
34 mins ago
$begingroup$
That's pretty good. A simple standard: at least one drivable road between neighbors
$endgroup$
– Will Jagy
29 mins ago
$begingroup$
just curious -- did you write code to do this, or did you do this by hand?
$endgroup$
– antkam
6 secs 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.
answered 1 hour ago
Misha LavrovMisha Lavrov
49.3k757108
$endgroup$
$begingroup$
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
35 mins 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
34 mins ago
$begingroup$
That's pretty good. A simple standard: at least one drivable road between neighbors
$endgroup$
– Will Jagy
29 mins ago
$begingroup$
just curious -- did you write code to do this, or did you do this by hand?
$endgroup$
– antkam
6 secs 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.
answered 1 hour ago
Misha LavrovMisha Lavrov
49.3k757108
$endgroup$
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.
answered 1 hour ago
Misha LavrovMisha Lavrov
49.3k757108
edited 26 mins ago
answered 1 hour ago
Misha LavrovMisha Lavrov
49.3k757108
answered 1 hour ago
Misha LavrovMisha Lavrov
49.3k757108
answered 1 hour ago
Misha LavrovMisha Lavrov
49.3k757108
49.3k757108
$begingroup$
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
35 mins 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
34 mins ago
$begingroup$
That's pretty good. A simple standard: at least one drivable road between neighbors
$endgroup$
– Will Jagy
29 mins ago
$begingroup$
just curious -- did you write code to do this, or did you do this by hand?
$endgroup$
– antkam
6 secs ago
add a comment |
$begingroup$
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
35 mins 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
34 mins ago
$begingroup$
That's pretty good. A simple standard: at least one drivable road between neighbors
$endgroup$
– Will Jagy
29 mins ago
$begingroup$
just curious -- did you write code to do this, or did you do this by hand?
$endgroup$
– antkam
6 secs ago
$begingroup$
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
35 mins ago
$begingroup$
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
35 mins 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
34 mins 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
34 mins ago
$begingroup$
That's pretty good. A simple standard: at least one drivable road between neighbors
$endgroup$
– Will Jagy
29 mins ago
$begingroup$
That's pretty good. A simple standard: at least one drivable road between neighbors
$endgroup$
– Will Jagy
29 mins ago
$begingroup$
just curious -- did you write code to do this, or did you do this by hand?
$endgroup$
– antkam
6 secs ago
$begingroup$
just curious -- did you write code to do this, or did you do this by hand?
$endgroup$
– antkam
6 secs ago
add a comment |
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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
1 hour 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
1 hour ago
$begingroup$
thank you for your suggestion, I made a few clarifications.
$endgroup$
– Bor Kari
55 mins ago