Graph Theory 8: Four Color Theorem (Kempe's Proof)

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The four color map theorem and Kempe's proof expressed in term of simple, planar graphs.

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Kempe’s approach is so intuitive! Thank you for sharing. You’ve helped me summarize the four color theorem for my high schoolers.

AndrewMarcell

your teaching style is amazing thanks a lot! I watched this video several times!

mahsarahimian

Thank you so much
You have made us to fall in love with graph theory

shilparudrannavar

Thank you so much.This helped me to prepare for the seminar named ' Kempe Chains'

akh

I was happy to find an explanation of the 5 color theorem in clear english!

georgelaing

Thank you for the video that highlights where mathematics did go wrong for once. I saw Kempe's proof in another video, and I had heard it was wrong, but I still needed a day to find a counterexample. Still a rewarding experience.

Achill

hi sir your counterexample either has a mistake or i'm missing something.

you forgot to connect the blue circle on the left to orange below it and blue circle on right to green below it at 16:08.

what this ended up doing is that blue circle on the left can simply be changed to orange without any harm (i'm not talking about alternating sequence just the one circle) and similarly the blue circle on the right can be changed to green without any harm.therefore eliminating both blue circles and blue color hence can be used to color that vertex.

this can be fixed by connecting blue circle on the left to orange below it and blue circle on the right to green below it.similarly green and orange circles at the bottom must be connected as well.

adamantiumcookieplayzz

Great video. You deserve much more views and likes.

bayezidx

Come on Math At Andrews, please try to answer my question: Is the max number of nodes with each-to-each connections in 2D (4) related to the minimum number of colors to separate map regions in 2D... Would calculating how many connectors cross for 5+ nodes help.. In 1D it's 2 nodes and 2 colors, in 3D, given infinite sized nodes/regions it's potentially infinite colors/connectors needed? (although finite, node+connector/region size-related formulas are computable).. I just want to know whether I'm totally barking up the wrong tree or on to something. I am a decent programmer, just not a super-advanced mathematician. I understand the official proof. We can share the prize! LOL.

PrivateSi

This video is great, thank you. Do you have a link to the source where you gathered this information on the four colour theorem? I would like to read more about this method.

No.plan.sam

Come on Math at Andrews, try and answer my question.. Is the max number of each-to-each connected objects without connectors crossing in 2D (4) related to the minimum number of colors needed to separate each region on a 2D map.. Would counting the number of crossed connector for more objects help with the proof... In 1D only 2 colors are needed to separate map regions and only 2 objects can be connected each to each.. In 3D the algos break down to infinity if we have infinite sized objects / regions.. Come on, help me out, we can share the prize if I'm correct!.. It's bugging me again. Put my mind at rest and tell me why I'm barking up the wrong tree, if I am... Cheers for highlighting the problem and solution, got me thinking.

PrivateSi

You should have explained why edges can't overlap in the real-world graphs or at least state that it was an exercise for the viewer. Also, you skipped the case when two adjacent colors are the same.

bayezidx

but isnt it flawed to assume that things can only connect based ona 2D plane? what if you didnt consider 2D or 3D planes? then what would happen???

christopherwalsh

The graph solution is much more complicated than mine... In 2D, the maximum number of nodes that can be connected to each other (each to each) without connectors crossing is 4... I know it's not an actual mathematical proof but it is a graphical abstraction of the same problem. I'd like to know if I'm correct, perhaps you could tell me, I don't know any proper mathematicians.

PrivateSi

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