The four-colour conjecture (4CC) is a question that asks whether any map can be coloured using only four colours, with the constraint that neighboring countries must have distinct colours. This conjecture has re- mained unanswered for over 170 years, with a rich history of various attempts to prove it. In 1879, Kempe put forward a proof, but it was invalidated by Heawood after 11 years. Heawood did succeed, however, in prov- ing the weaker five-colour theorem. It wasn’t until 1976 that the first genuine proof was discovered by Appel and Haken. This proof sparked controversy due to its reliance on approximately 1200 hours of computer computation, making it unverifiable by hand.
The purpose of this report is to provide a comprehensive overview of the historical background and re- cent advancements in the four-colour theorem. It will delve into the numerous failed attempts to prove the conjecture and discuss the groundbreaking proof by Appel and Haken. Additionally, it will explore a recent endeavor by Dr. Xiang, who approached the problem from a different perspective but also encountered a fallacy in his work. The main contribution of this thesis involves a new proof that builds upon the research of Dr. Yeh, who attempted to prove the four-colour theorem by transforming it into a system of linear equa- tions. The missing crucial steps will be addressed for whose correction new ideas will be introduced, using an integral version of Farkas’ lemma and superadditivity. The ideas being researched do not finalize the proof but contribute towards a possible elegant proof in the future. Finally, related topics such as maps on different surfaces and maps with disconnected regions will be considered, broadening the scope of the discussion.