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Register a new userA Study on Hand Proof for The Four-Color Theorem
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For the four-color theorem that has been developed over one and half centuries, all people believe it right but without complete proof convincing all1-3. Former proofs are to find the basic four-colorable patterns on a planar graph to reduce a map coloring4-6, but the unavoidable set is almost limitless and required recoloring hardly implements by hand7-14. Another idea belongs to formal proof limited to logical operation15. However, recoloring or formal proof way may block people from discovering the inherent essence of a coloring graph. Defining creation and annihilation operations, we show that four colors are sufficient to color a map and how to color it. We find what trapped vertices and boundary-vertices are, and how they decide how many colors to be required in coloring arbitrary maps. We reveal that there is the fourth color for new adding vertex differing from any three coloring vertices in creation operation. To implement a coloring map, we also demonstrate how to color an arbitrary map by iteratively using creation and annihilation operations. We hope our hand proof is beneficial to understand the mechanisms of the four-color theorem.
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For a given graph $G(V,E)$ and one of its dominating set $S$, the subgraph $Gleft[Sright]$ induced by $S$ is a called a dominating tree if $Gleft[Sright]$ is a tree. Not all graphs has a dominating tree, we will show that a graph without cut vertices has at least one dominating tree. Analogously, if $Gleft[Sright]$ is a forest, then it is called a dominating forest. As special structures of graphs, dominating tree and dominating forest have many interesting application, and we will focus on its application on the problem of planar graph coloring.
In [J. Combin. Theory Ser. B 70 (1997), 2-44] we gave a simplified proof of the Four-Color Theorem. The proof is computer-assisted in the sense that for two lemmas in the article we did not give proofs, and instead asserted that we have verified those statements using a computer. Here we give additional details for one of those lemmas, and we include the original computer programs and data as ancillary files accompanying this submission.
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 between any two vertices. Based on ECT, a simple proof for four color theorem for planar graph is then proposed.
The purpose of these notes is to present a fairly complete proof of the classification Theorem for compact surfaces. Other presentations are often quite informal (see the references in Chapter V) and we have tried to be more rigorous. Our main source of inspiration is the beautiful book on Riemann Surfaces by Ahlfors and Sario. However, Ahlfors and Sarios presentation is very formal and quite compact. As a result, uninitiated readers will probably have a hard time reading this book. Our goal is to help the reader reach the top of the mountain and help him not to get lost or discouraged too early. This is not an easy task! We provide quite a bit of topological background material and the basic facts of algebraic topology needed for understanding how the proof goes, with more than an impressionistic feeling. We hope that these notes will be helpful to readers interested in geometry, and who still believe in the rewards of serious hiking!
We show that the mathematical proof of the four color theorem yields a perfect interpretation of the Standard Model of particle physics. The steps of the proof enable us to construct the t-Riemann surface and particle frame which forms the gauge. We specify well-defined rules to match the Standard Model in a one-to-one correspondence with the topological and algebraic structure of the particle frame. This correspondence is exact - it only allows the particles and force fields to have the observable properties of the Standard Model, giving us a Grand Unified Theory. In this paper, we concentrate on explicitly specifying the quarks, gauge vector bosons, the Standard Model scalar Higgs $H^{0}$ boson and the weak force field. Using all the specifications of our mathematical model, we show how to calculate the values of the Weinberg and Cabibbo angles on the particle frame. Finally, we present our prediction of the Higgs $H^{0}$ boson mass $M_{H^{0}} = 125.992 simeq 126 GeV$, as a direct consequence of the proof of the four color theorem.
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