Related papers: A manual proof of the Four-colour Theorem
We give a new proof of Lucas' Theorem in elementary number theory.
We present simple and direct proof to an important case of Nash-Moser-Ekeland theorem.
This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…
In this paper, we apply an equivalent color transform (ECT) for a minimal $k$-coloring of any graph $G$. It contracts each color class of the graph to a single vertex and produces a complete graph $K_k$ for $G$ by removing redundant edges…
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…
We present a new proof of the known fact that the chromatic number of the plane is at least 5. The main difference of this proof is that it can be verified manually without the help of the computer.
In this note we give a detailed proof of a theorem of Aubin.
This paper provides a new simple proof of Hesse's theorem in projective geometry for any dimension.
We interpret the number of good four-colourings of the faces of a trivalent, spherical polyhedron as the 2-holonomy of the 2-connection of a fibered category, phi, modeled on Rep(sl(2)) and defined over the dual triangulation, T. We also…
We give a simple proof of the Fourier Inversion Theorem, using the methods of nonstandard analysis.
Standard proofs of Lusin's theorem, using simple functions, are sometimes quite elaborate. Here, we give a one-sentence proof of Lusin's theorem. We do not believe our approach, by way of inverse images, is new. However, this particular…
We give a concise proof of the fundamental theorem of smoothing theory in the special case when a smoothing exists.
We present a simple short proof of the Fundamental Theorem of Algebra, without complex analysis and with a minimal use of topology. It can be taught in a first year calculus class.
This appendix for our article, "Almost-rainbow edge-colorings of some small subgraphs", contains the full proof of Theorem 4.1.
This paper is a collection of thoughts and observations, being partly a review and partly a report of current research, on recent work in various aspects of Gr\"unbaum colorings, their existence and usage. In particular, one of the most…
We show, without using the Four Color Theorem, that for each planar triangulation, the number of its proper vertex colorings by 4 colors is a determinant and thus can be calculated in a polynomial time. In particular, we can efficiently…
We give a simple direct proof of Fermat's two squares theorem. Our argument uses no intricate notions or ideas; one might say that it is a proof by careful bookkeeping. As such, the proof may be particularly easy to comprehend by students…
The attempts to prove the Four Color Problem last for long years. A little hope arises that the properties of the minimal partial triangulations will be very useful for the solution of the Four Color Problem. That is why the material of…
We prove the termination of 4-fold canonical flips.
The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.