Since 1850, the four-color theorem has become one of the most controversial problems in the history of mathematics and one of the "three major problems in modern mathematics in the world" because of its "intuitive simplicity but difficult to prove". In this paper, we use the graph theory to prove the four-color theorem by the unified construction, decomposition and regular splicing of the planar graph. Firstly, based on the theory of maximal planar graph, this paper decomposes the maximal planar graph through the graph surrounding coloring model, and constructs the adjacency relationship. This process not only preserves the original adjacency property of planar graphs, but also provides a basis for the subsequent coloring relations. Based on these decompositions, the four-colorability of the graph surrounding coloring model is further proved by using the number line abstract relation that can be logically understood. Then, according to the adjacency relationship of the surrounding shading models, the three-dimensional number axis shading model is constructed, and the adjacency relationship is simplified. This simplification not only reduces the complexity of coloring combinations, but also makes the analysis of adjacency coloring relations more intuitive. Based on the simplified coloring combination, this paper re-colors the odd and even cycles surrounding the adjacency of the coloring model, and further simplifies the combination of triangular adjacency coloring relations through the equivalent substitution of the values of the adjacent endpoints on the three-dimensional number axis. On the basis of the above simplified combination, through a series of proof steps, we verify the four-colorability of planar graphs under various combinations. These steps include, but are not limited to, an extension of the adjacency distance around the module based on the conclusions demonstrated above, an extension of the adjacency case outside the triangular adjacency window, and an extension by a regular combination of odd and even adjacency end cycles (OAEC and EAEC). By these extensions, the conclusion is proved to cover the whole planar graph. Finally, this paper proposes a logical proof method for the four-color theorem, which simplifies the complexity of the coloring relation through a series of graph transformation rules, and extends the rules effectively, and finally provides a proof of the four-color theorem for finite planar graphs. This method not only shows the potential of graph theory logic in solving complex mathematical problems, but also provides a new perspective and tool for the corresponding field research.

Peng Banghuang. The Proof of The Four Color Conjecture Based on Graph Theory Logic. Academic Journal of Mathematical Sciences (2024) Vol. 5, Issue 3: 91-129. https://doi.org/10.25236/AJMS.2024.050313.

[1] Liu Daoyu. The origin, function and crisis of mathematical conjecture [J]. Exploration of Higher Education, 2022, No.3, 5-7 

[2] Wilson, R. J. Four colors suffice: How the map problem was solved.[M].Allen Lane Science.2002

[3] Biggs, N. L., Lloyd, E. K., & Wilson, R. J..Graph Theory, 1736-1936[M].Oxford:Clarendon Press. 1986

[4] Robin, W. J. The history of the four color theorem.[J].American Mathematical Monthly, 113(4), 318-326. 2006 

[5] Demaine, E. D., & Hearn, R. W. Playing games with algorithms: Algorithmic combinatorial game theory. [J]. Foundations and Trends in Theoretical Computer Science, 4(1), 1-86. 2009

[6] Dirac G A. Percy John Heawood [J]. J. London Math. Soc. 1963, 38: 263-277

[7] Kempe, A. B. On the geographical problem of the four colours. [J]. Proceedings of the London Mathematical Society, s1-10(1), 75-83. 1879

[8] Heawood, P. J. Map colour theorem. [J]. Proceedings of the London Mathematical Society, s2-24(1), 336-338. 1890

[9] Appel, K., & Haken, W. Every planar map is four colorable. Part I: Discharging. [J]. Illinois Journal of Mathematics, 21(3), 429-490. 1977

[10] Appel, K., Haken, W., & Koch, J. Every planar map is four colorable. Part II: Reducibility. [J]. Illinois Journal of Mathematics, 21(3), 491-567. 1977

[11] Robertson, N., Sanders, D. P., Seymour, P., & Thomas, R. The four-colour theorem. [J]. Journal of Combinatorial Theory, Series B, 70(1), 2-44. 1997

[12] Gonthier, G. Formal proof—the four-color theorem. [J]. Notices of the American Mathematical Society, 55(11), 1382-1393. 2008

[13] Wang Xianfen, Hu Zuoxuan. Three-generation proof of the four-color theorem [J]. Dialectics of Nature Newsletter, 2010, 32 (1): 42-48

[14] Liang Fang. Reflection on mathematical philosophy caused by computer [D]. Graduate School of Chinese Academy of Social Sciences, 2002, No.01

[15] Yang Jun, Li Gaoping, Li Qing. A brief comment on a non-computer "logical proof" of the four-color theorem [J]. Journal of Southwest Minzu University (Natural Science Edition) 2021, Vol. 47, No.3 

[16] Bondy, J. A., & Murty, U. S. R. Graph Theory.[M]Springer. 2008

[17] WEST D B. Introduction to graph theory [M]. Version 2. Translated by Li Jianzhong, Luo Jizhou. Beijing: China Machine Press, 2020

[18] Xie Litong, Liu Guizhen. On the conjecture of whitney and tutte [J]. Acta Mathematica Sinica, 1995, Vol.38 No.3

[19] Zhang Mengran, Four color theorem used to analyze the magnetic properties of crystals [J]. Science and Technology Daily, 2014, Edition 001

[20] Mohar, B., & Thomassen, C. Graphs on surfaces.[M]Johns Hopkins University Press. 2001

[21] Deng Shuo, Wang Xianfen, The process of proving the four-color theorem and its influence on graph theory [J]. Science and Technology Horizon 2095-2457.2016.25.089

[22] Xu Jin, Li Zepeng, Zhu Enqiang. Advances in the theory of maximal planar graphs [J].Journal of Computer Science, 2015, 38 (8): 1680-1704 

[23] Pach, J., & Wenger, R. Embedding planar graphs at fixed vertex locations.[J].Graphs and Combinatorics, 12(4), 319-326. 1996

[24] Ye Zailiang, Zhang Yanmin, A class of four-colorable maximal planar graphs [J]. Journal of Shangluo Teachers College, 2000, Vol. 14, No.4 

[25] Diestel, R. Graph Theory.[M]Springer 2010

[26] You Chengye. Lectures on basic topology [M]. Beijing: Peking University Press, 2006

[27] Tutte, W.T. "Graph theory as I have known it." [M] Oxford University Press. 1998

[28] Thomas, R. "An update on the four-color theorem."[J]Notices of the American Mathematical Society, 45(7), 848-859. 1998