or containing a vertex that touches five other vertices, called its neighbors. As such, to prove the four color theorem, it is sufficient to prove that vertices of five or less were all four-colorable. Kempe was able to prove the case of degree four and give a partial proof of degree five using Kempe chains. In this case, Kempe chains are used to prove the idea that no vertex has to be touching four colors different from itself, i.e., having a degree of 4. First, one can create a graph with a vertex v and four vertices as neighbors. If we remove the vertex v, we can four-color the remaining vertices. We can set the colors as (in clockwise order) red, yellow, blue, and green. In this situation, there can be a Kempe chain joining the red and blue neighbors or a Kempe chain joining the green and yellow neighbors, but not both, since these two paths would necessarily intersect, and the vertex where they intersect cannot be colored. Supposing that the Kempe chain is connecting the green and yellow neighbors, red and blue must then necessarily not have a Kempe chain between them. So, when placing the original vertex v back into the graph, we can simply reverse the colors of the red vertex and its neighbors (including the red vertex, making it blue) and color vertex v as red. This results in a four-colored graph. Other Applications Kempe chains have been used to solve problems in coloring extension. See also Four color theorem Five color theorem Graph coloring References Graph coloring