Related papers: A mathematical proof of four color theorem
This paper has been withdrawn due to a crucial theoretical error.
This paper has been withdrawn by the author due to a crucial sign error in equation 1
This paper has been withdrawn by the author.
This paper has been withdrawn by the author(s) in the light of several other works available and due to a misunderstanding in the authorships.
This paper has been withdrawn by the authors, due a crucial error in the proof of the main theorem.
We give a pictorial proof that transparently illustrates why four colours suffce to chromatically differentiate any set of contiguous, simply connected and bounded, planar spaces; by showing that there is no minimal planar map. We show,…
This paper has been withdrawn by the authors. We have discovered an error in the evaluation of the diagram, which invalidates our conclusion.
We give a simple reformulation of the four color theorem as a problem on strings over a four letter alphabet.
This paper has been withdrawn by the author due to a crucial error.
This paper has been withdrawn by the author due to a crucial sign error.
This paper has been withdrawn by the author, due an error in the proof of Proposion 2.13.
The famous four color theorem states that for all planar graphs, every vertex can be assigned one of 4 colors such that no two adjacent vertices receive the same color. Since Francis Guthrie first conjectured it in 1852, it is until 1976…
This paper has been withdrawn by the author due to a crucial sign error in Proposition 3.1.
This paper has been withdrawn by the authors. We have discovered an error in the evaluation of the diagram, which invalidates our conclusion.
This paper has been withdrawn by the author, due to errors in the figures.
In 1976, Appel and Haken achieved a major break through by proving the four color theorem $(4CT)$. Their proof is based on studying a large number of cases for which a computer-assisted search for hours is required. In 1997, Robertson,…
This paper has been withdrawn by the author due to the incorrect argument for the security.
This paper has been withdrawn.
This paper has been withdrawn by the author due to some errors.
This paper has been withdrawn by the author due to an error in an inequality in the proof of Theorem 1.1.