Attempts and Inferences of the Four-Color Theorem

Abstract

The Four-Color Theorem is a classic problem in graph theory, stating that any planar map can be colored using no more than four colors so that no adjacent regions share the same color. Since 1976, when Appel and Haken used computer assistance to prove this theorem, it has been considered solved. However, due to its complexity and the difficulty of manually verifying the proof, some mathematicians still have doubts. This study proposes a new logical approach to provide an alternative proof for the Four-Color Theorem. Using a combination of theoretical derivations and graph theory tools, the paper first analyzes the basic structures of planar graphs, then use inductive reasoning to confirm coloring patterns in small graphs, gradually extending the method to more complex graphs. It developed a specific algorithm that simplifies vertices and edges in planar graphs, showing that any complex planar graph can be reduced to a few basic shapes, allowing it to be colored with fewer than four colors. Our results show that this reasoning model can confirm the Four-Color Theorem for a range of planar graphs without computer assistance. Though not covering all graph types, this model offers new insights for future research, especially in simplifying large graphs.