Four Color Theorem is equivalent to the statement: "Every cubic planar bridgeless graphs is 3-edge colorable". There is computer assisted proof given by Appel and Haken. Dick Lipton in of his [beautiful blogs](http://rjlipton.wordpress.com/2009/04/24/the-four-color-theorem/) posed the following open problem:



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> Are there non-computer based proofs of the Four Color Theorem?
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Surprisingly, While I was reading this paper, 
Anshelevich and Karagiozova, [Terminal backup, 3D matching, and covering cubic graphs](http://portal.acm.org/citation.cfm?id=1250790.1250849), the authors state that Cahit proved that "every 2-connected cubic planar graph is edge-3-colorable" which is equivalent to the Four Color Theorem (I. Cahit, Spiral Chains: The Proofs of Tait's and Tutte's Three-Edge-Coloring Conjectures. arXiv preprint, math CO/0507127 v1, July 6, 2005).



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> Does Cahit's proof resolve the open problem in Lipton's blog by providing non-computer based proof for the Four Color Theorem?
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Cross posted on math.stackexchange.com as [Human checkable proof of the Four Color Theorem?](https://math.stackexchange.com/questions/8752/human-checkable-proof-of-the-four-color-theorem)



Human checkable proof of the Four Color Theorem?
This is too long for a comment, so I am placing it here.


In this [article of the Notices of the AMS](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.141.714&rep=rep1&type=pdf), Gonthier describes a full formal proof of the four-color theorem, which makes explicit every logical step of the proof. 


Although this formal proof has been checked by the [Coq proof system](http://coq.inria.fr/), it would seem to be a category error to view this proof as a computer-based proof of the same kind as Appel and Haken's. The situation with Gonthier's proof is that we essentially have a full written text constituting a verified formal proof of the four-color theorem in first order logic.


And that is a state of certainty that most theorems in mathematics, including many of the classical results, have not yet attained.


