In this note we are proving the four conjecture for planar graphs. The proof&nbsp; follows the Euler Indentity for planar graphs

Czopik, Janusz

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Journal of Progressive Research in MathematicsVol 18,2 - 2021 — E-ISSN: 2395-0218http://www.scitecresearch.com/journals/index.php/jprmThe remarks on the four color problemJanusz Czopik2715 Cane Field Dr Sugar Land TX 77479U. S. A..Received: February 26, 2021; Accepted: March 29, 2021; Published: June 15, 2021Copyright © 2021 by author(s) and Scitech Research Organisation(SRO).This work is licensed under the Creative Commons Attribution International License (CC BY).http://creativecommons.org/licenses/by/4.0/AbstractThe proof follows the Euler Identity for planar graphs. In this note we are proving the fourcolor conjecture for planar graphs.The proof follows the Euler Identity for planar graphs.KeywordsFour color conjecture, planar graphs.1 IntroductionWe will concern notoriously difficult, the four color problem forplanar graphs.A graph G is a planar graph if G can be drawn in the plane sothat no two of its edges cross. Let G be a connected planargraph with n−vertices, m−edges and r- regions.The four color conjecture, famous from 1860: the regions of everyplanar graph can be colored with four or fewer colors in such away that every two regions sharing a common boundary boundaryare colored differently [ 1 ].For many years unsolved, the four color problem was attakedunsuccessfully by very famous mathematicians [ 1 ], [2].Finally in 1976 two americans mathematicians: Kenneth Appeland Wolfgang Hake gave a proof. The proof is highly controversialbecause related on computer. .In 1996 group of four Americans mathematicians: Neil Robertson,Daniel Sanders, Paul Seymor and Robin Thomas relied heavilyon computer and used the same approach gave own proof.Cite this paper: Janusz Czopik (2021). The remarks on the four color problem. Journal of ProgressiveResearch in Mathematics, 18(2), 59-60. Retrieved fromhttp://scitecresearch.com/journals/index.php/jprm/article/view/20301Janusz CzopikAmazingly, each of the proofs required different number of theso called ”unavoidable configurations”. What does it means itis unclear. The question remaining, are there real proofs offour colors conjecture?2. Proof of the four color conjecture.If G is a connected planar graph with n−vertices, m−edges andr−regions then(1) n−m + r = 2 (the Euler Identity for planar graphs [ 1])We rewrite (1) in the form(2) r = 2− n + m.We will ask about the minimum of (2) under following assumptions(3) n ≥ 3(4) m ≤ 6(5) m ≤ 3n− 6The assumption (3) means, we consider the graphs of minimum3 vertices.The assumption (4) means, we consider the planar graphs ofmaximum 6 edges.Though, the maximal planar grapth contains9 edges, the same numbers of edges contains non planar graphK3,3.The assumption (5) is one of the theorem of the planar graphs [1 ].The system of inequalities (3)→(5) gives us as solution only onepoint (4,6). This means that r = 4.3. ConclusionBecause theorem (5): m ≤ 3n− 6 is proven using the EulerIdentity we see that the proof of four color theorem real is basedon the Euler Identity.References[1] Arthur Benjamin, Gary Chartrand, Ping Zhung, The fascinatingword of graph theory, Princeton University Press, 2015.[2] en. wikipedia.org/wiki/four color theorem60

The remarks on the four colors problem

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The remarks on the four colors problem

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