A Kempe chain in a colored graph is a maximal connected component containing at most two colors. Kempe chains have played an important role historically in the study of the Four Color Problem. Some methods of systematically applying Kempe chain color exchanges have been studied by Alfred Errera and Weiguo Xie. A map constructed by Errera represents an important counterexample to some implementations of these methods. Using the ideas of Irving Kittell, we determine all colorings of the Errera map which form such a counterexample and describe how to color them individually. We then extend our results from the Errera map to a family of graphs containing the Errera map in a specific way. Being able to color this family of graphs appears to address many cases which prove difficult for the previous systematic color exchange methods.

The four-color problem was first posed by Francis Guthrie in 1852. Over a century, many researchers tried many ways and obtained some useful results. One proposed proof was given by Alfred Kempe using Kempe Chain in 1879, but Percy Heawood found counterexample of Kempe's proof in 1890 [3]. This historic Heawood map has 25 regions. It can be very challenging to just use trial and error method to make it four-colored. In this paper, a systematic way to color the map with four colors using a novel method of rotation [4] inspired by a rotation principle from Zhuan Falun book [2] of Falun Dafa will be demonstrated. It shows that the novel method of rotation is very powerful and can provide a systematic approach to make maps four-colored.

The four-color problem was first posed by Francis Guthrie in 1852. It remained unsolved for over a century. Although computer-assisted proofs emerged in last 50 years, they were somewhat the "machine-checkable proof", which were hardly checked by human readers. In this paper, a novel method of rotation inspired by a rotation principle from Zhuan Falun book of Falun Dafa has been developed to prove the Four Color Theorem.