The five color theorem is a result from graph theory that given a plane separated into regions, such as a political map of the countries of the world, the regions may be colored using no more than five colors in such a way that no two adjacent regions receive the same color. The five color theorem is implied by the stronger four color theorem, but is considerably easier to prove. It was based on a failed attempt at the four color proof by Alfred Kempe in 1879. Percy John Heawood found an error 11 years later, and proved the five color theorem based on Kempe's work.

Outline of the proof by contradiction

First of all, one associates a simple planar graph G to the given map, namely one puts a vertex in each region of the map, then connects two vertices with an edge if and only if the corresponding regions share a common border. The problem is then translated into a graph coloring problem: one has to paint the vertices of the graph so that no edge has endpoints of the same color. Because G is a simple planar graph, i.e. it may be embedded in the plane without intersecting edges, and it does not have two vertices sharing more than one edge, and it does not have loops, then it can be shown (using the Euler characteristic of the plane) that it must have a vertex shared by at most five edges. (Note: This is the only place where the five-color condition is used in the proof. If this technique is used to prove the four-color theorem, it will fail on this step. In fact, an icosahedral graph is 5-regular and planar, and thus does not have a vertex shared by at most four edges.) Find such a vertex, and call it v. Now remove v from G. The graph G' obtained this way has one fewer vertex than G, so we can assume by induction that it can be colored with only five colors. If the coloring did not use all five colors on the five neighboring vertices of v, it can be colored in G with a color not used by the neighbors. So now look at those five vertices v_1, v_2, v_3, v_4, v_5 that were adjacent to v in cyclic order (which depends on how we write G). So we can assume that v_1, v_2, v_3, v_4, v_5 are colored with colors 1, 2, 3, 4, 5 respectively. Now consider the subgraph G_{1,3} of G' consisting of the vertices that are colored with colors 1 and 3 only and the edges connecting them. To be clear, each edge connects a color 1 vertex to a color 3 vertex (this is called a Kempe chain). If v_1 and v_3 lie in different connected components of G_{1,3}, we can swap the 1 and 3 colors on the component containing v_1 without affecting the coloring of the rest of G'. This frees color 1 for v completing the task. If on the contrary v_1 and v_3 lie in the same connected component of G_{1,3}, we can find a path in G_{1,3} joining them that consists of only color 1 and 3 vertices. Now turn to the subgraph G_{2,4} of G' consisting of the vertices that are colored with colors 2 and 4 only and the edges connecting them, and apply the same arguments as before. Then either we are able to reverse the 2-4 coloration on the subgraph of G_{2,4} containing v_2 and paint v color 2, or we can connect v_2 and v_4 with a path that consists of only color 2 and 4 vertices. Such a path would intersect the 1-3 colored path we constructed before since v_1 through v_5 were in cyclic order. This is clearly absurd as it contradicts the planarity of the graph. So G can in fact be five-colored, contrary to the initial presumption.

Linear time five-coloring algorithm

Multiple authors, beginning with Lipton and Miller in 1978, have studied efficient algorithms for five-coloring planar graphs. The algorithm of Lipton and Miller took time O(n\log n), but subsequent researchers reduced the time bound to O(n). The version below is from a 1996 paper by Robertson, Sanders, Seymour, and Thomas, which describes it briefly in connection with a slower O(n^2)-time algorithm for four-coloring. The algorithm as described here operates on multigraphs and relies on the ability to have multiple copies of edges between a single pair of vertices. It is based on Wernicke's theorem, which states the following: We will use a representation of the graph in which each vertex maintains a circular linked list of adjacent vertices, in clockwise planar order. In concept, the algorithm is recursive, reducing the graph to a smaller graph with one less vertex, five-coloring that graph, and then using that coloring to determine a coloring for the larger graph in constant time. In practice, rather than maintain an explicit graph representation for each reduced graph, we will remove vertices from the graph as we go, adding them to a stack, then color them as we pop them back off the stack at the end. We will maintain three stacks: The algorithm works as follows:

Alternate proof

Kainen (1974) provides a simplified proof of the five color theorem, based on the non-planarity of K6 (the complete graph with 6 vertices) and graph minors. This proof generalizes to graphs that can be made planar by deleting 2 edges.