The known proof of the 4CT is to show that there is no minimal counter-example (no map that require five colors). But this approach is like searching a finite number cases within of huge number of maps (triangulated planar graphs). Furthermore for each configuaration a coloring algorithm must be employed to show that they are 4-colorable. Certainly this is not a constructive proof. The constructive proof which I have proposed is that you devise an coloring algorithm based on the property of the maximal planar graph (here property is the nested spiral chains). Then backward coloring of these spiral chains and showing that no more than four colors are required for any maximal planar graph. In the game of Rubik's cube the 25 moves is an attained upper bound like to say 5 colors is suffice to color any map (elegant proof is an old result of Heawood). That is why the next open case for this problem is 24 moves. In order to make the problem more difficult I would ask if there is constructive (algorithmic) proof that uses 24 moves or less for any configuration?
My comment is about the four color theorem. Yes indeed Appel&Haken,1976 and Robertson et.al. 1996 proofs are computer assisted. There is a belief that this theorem cannot be settled without a huge computational effort. This is wrong. I have proposed a non-computer proof in 2004 (see arXiv:math/0408247v1 and arXiv:0710.2066v2 ) which cuts or bypass computational cases and proves the theorem algorithmically by using spiral chain coloring. That is no matter what is the shape of the map e.g., the maximal planar graph the algorithm colors regions (nodes) with at most four colors. I have already received some skepticism about my proof e.g., there must be something wrong here but we can't show or similar. But these are irrelevant. In fact the known proofs (which I beleive are correct) still attract critics from the point of huge computer computations.
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The known proof of the 4CT is to show that there is no minimal counter-example (no map that require five colors). But this approach is like searching a finite number cases within of huge number of maps (triangulated planar graphs). Furthermore for each configuaration a coloring algorithm must be employed to show that they are 4-colorable. Certainly this is not a constructive proof. The constructive proof which I have proposed is that you devise an coloring algorithm based on the property of the maximal planar graph (here property is the nested spiral chains). Then backward coloring of these spiral chains and showing that no more than four colors are required for any maximal planar graph. In the game of Rubik's cube the 25 moves is an attained upper bound like to say 5 colors is suffice to color any map (elegant proof is an old result of Heawood). That is why the next open case for this problem is 24 moves. In order to make the problem more difficult I would ask if there is constructive (algorithmic) proof that uses 24 moves or less for any configuration?
My comment is about the four color theorem. Yes indeed Appel&Haken,1976 and Robertson et.al. 1996 proofs are computer assisted. There is a belief that this theorem cannot be settled without a huge computational effort. This is wrong. I have proposed a non-computer proof in 2004 (see arXiv:math/0408247v1 and arXiv:0710.2066v2 ) which cuts or bypass computational cases and proves the theorem algorithmically by using spiral chain coloring. That is no matter what is the shape of the map e.g., the maximal planar graph the algorithm colors regions (nodes) with at most four colors. I have already received some skepticism about my proof e.g., there must be something wrong here but we can't show or similar. But these are irrelevant. In fact the known proofs (which I beleive are correct) still attract critics from the point of huge computer computations.