Real MATHEMATICS – Graphs #6

Bequeath Problem

King Serkan I decides to allocate his lands to his children. Obviously he had set up some ground rules for the allocation:

  • Each child will get at least one land.
  • Same child can’t have adjacent lands.

Problem: At least how many children should Serkan I has so that allocation can be done without a problem?

Map #1

Let’s start from a simple map:

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In this case Serkan I can have two children:

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As you can remember from the previous article a map and a graph is irreversible. If we represent lands with dots, and let two dots be connected with a line if they are adjacent, we can show maps as graphs:

20190423_000334.jpg

Let’s add another land to this map:

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Adding a land on a map is the same thing as adding a dot on a graph:

20190423_000507.jpg

Map #2

Let’s assume there are three lands on a map:

20190423_000523.jpg

We can convert this map into graph as follows:

20190423_000537.jpg

As seen above, three children are needed in order to fulfill Serkan I’s rules:

20190423_000553.jpg

Map #3

Let the third map be the following:

20190423_000627.jpg

According to Serkan I’s rules, we will need four children for such map:

20190423_000642.jpg

Map #3 can be shown as a graph like the following:

20190423_000728.jpg

Map #4

For the final map, let’s assume Serkan I left a map that looks like USA’s map:

480271690e1e0485f71988e273730559

Surprisingly four children are enough in order to allocate the lands on the map of USA:

amarikaaa

What is going on?

Careful readers already noticed that adding a dot that connects the other dots in the second map’s graph gives the graph of the third map. Same thing is true for the first and second maps:

Hence, adding a new dot to the graph means adding a new child.

Q: Is it possible to create a map that requires at least five children?

In other words: Is it possible to add a fifth dot to the graph so that it has connection to all existing four dots? (Ps: There can be no crossing in a graph as our maps are planar.)

Then, all we have to do is to add that fifth dot… Nevertheless I can’t seem to do it. When I add the fifth dot outside of the following graph:

It is impossible to connect 1 and 5 without crossing another line. No matter what I try, I can’t do it:

Four Color Theorem

About 160 years ago Francis Guthrie was thinking about coloring maps:

“Can the areas on any map be colored with at most four colors such that no pair of neighboring areas get the same color?”

Incredibly the answer is yes.

This simple problem was introduced for the first time by Francis Guthrie in 1852. Not until 1976 there was no proof for Guthrie’s conjecture. Only then with the help of computers the conjecture was proved. This proof is crucial for mathematics world as it is known as the mathematical theorem that was proven with the help of computers.

One wonders…

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Add a fourth dot to the graph you see above and connect that dot to the existing dots. (You are free to place the fourth dot wherever you want on the graph.)

Now check your graph: One of those four dots is trapped inside the lines, isn’t it?

Can you fix that?

Explain how you can/can’t do it.

M. Serkan Kalaycıoğlu

Real MATHEMATICS – Graphs #5

Cruel Traffic Light

I have to drive past the same crossroad almost every single day. Obviously I stop at the longest-lasting traffic light of the crossroad. Within time I started loving these moments because it gave me a chance to think my life over. Though, it doesn’t take too long for me to start thinking about mathematics.

One of those days I found myself questioning the traffic lights and their relationship with mathematics. (After all mathematics is everywhere; isn’t it?) Soon after I realized that there was my beloved graph theory behind traffic lights.

Light #1

Let’s assume a one-way street with two traffic lights: One for the vehicles (we’ll call it A), other for the pedestrians (we’ll call it B):

20190419_014641.jpg

In such situation we have to avoid accidents. This means whenever A has green light, B must have red light and vice versa. We will not take account of the situation when both lights are red. Because even though there won’t be any accident, neither of the sides will be standing still (which is nonsense):

20190419_014737.jpg

All these can be shown using graph theory: Lights will be represented by dots. Dots in a graph will be connected with lines if they are not in the same color:

20190419_014752.jpg

Same thing can be shown with maps: If A and B are two neighboring countries, they should be colored in different colors to avoid confusion:

20190419_014817.jpg

Light #2

This time we will assume a two-way street with three traffic lights: Two for the vehicles from opposite sides (we’ll call them A and B), and one for the pedestrians (we’ll call that C):

20190419_014838.jpg

In such a situation when C is red, either or both of A and B should be green. When C is green, then both A and B should be red:

20190419_014912.jpg

We’ll skip the situation where all three of them are red as no one would move in such situation.

We can show these using graph theory and map coloring as follows:

20190419_014930.jpg

Since we set all the rules graphs make it much easier and clearer to understand the situations.

Light #3

Finally we have a two-way street (A and B), a right turn (C) and two pedestrian lights (D and E) as follows:

20190419_014947.jpg

This time it is a much complex situation:

20190419_015030.jpg

Although using graph theory makes it all easier for us to understand:

How Many Colors?

We will color the following graphs using the same rules we just established above: If two dots are connected with a line, then those dots must have different colors.

Chromatic Numbers: Whenever a graph is being colored, ambition is the use the least number of colors. This number is also known as the chromatic number of a graph.

Here we need 3 colors. Hence chromatic number of the graph is 3. Let’s add one more dot and line to the graph:

This time chromatic number of the graph becomes 2. We’ll add another dot and line:

Now chromatic number becomes 3 again. Let’s add a dot and a line for the last time:

Chromatic number is back to 2.

To be continued…

One Wonders…

What did just happen? What did you notice? Why is it happening?

How can you increase the chromatic number?

M. Serkan Kalaycıoğlu