The five color theorem is a result from graph theory that given a plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than five colors in such a way that no two adjacent regions receive the same color. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects

The five color theorem is implied by the stronger four color theorem, but can be proven with considerably less effort. The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country It was based on a failed attempt at the four color proof by Alfred Kempe in 1879. Sir Alfred Bray Kempe DCL FRS (6 July 1849 Kensington, London – 21 April 1922 London) was a Mathematician Percy John Heawood found a fatal error 11 years later, and proved the five color theorem based on Kempe's work. Percy John Heawood ( 8 September, 1861 Newport Shropshire, England - 24 January, 1955 Durham, England The four color theorem was finally proven by Kenneth Appel and Wolfgang Haken at the University of Illinois, with the aid of a computer. Kenneth Appel (born 1932 is a mathematician who in 1976 with colleague Wolfgang Haken at the University of Illinois at Urbana-Champaign, solved one of the most famous This article is about the flagship campus For other uses and locations of University of Illinois, see University of Illinois (disambiguation The University of A computer-assisted proof is a Mathematical proof that has been at least partially generated by computer They were assisted in some algorithmic work by John Koch. John Koch (1909-1978 is an American painter One of his works is The Sculptor (1964 an oil on canvas painting

Contents 1 Outline of the proof by contradiction 2 Linear time five-coloring algorithm 3 See also 4 References

Outline of the proof by contradiction

First of all, one associates a 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. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. 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.

The proof relies on the Euler characteristic to show that there must be a vertex V shared by at most five edges, and on the fact that G is planar, i. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant Kuratowski's and Wagner's theorems The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of Forbidden e. it may be embedded in the plane without intersecting edges.

Now remove V from G. The graph V' obtained this way has fewer vertices than G, so we can assume by induction that it can be colored with only five colors. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that V must be connected to five other vertices, since if not it can be colored in G with a color not used by them. So now look at those five vertices V₁, V₂, V₃, V₄, V₅ that were adjacent to V in cyclic order (which depends on how we write G). If we did not use all the five colors on them, then obviously we can paint V in a consistent way to render our graph 5-colored.

So we can assume that V₁, V₂, V₃, V₄, V₅ are colored with colors 1, 2, 3, 4, 5 respectively.

Now consider the subgraph G₁₃ of V' consisting of the vertices that are colored with colors 1 and 3 only, and edges connecting two of them. If V₁ and V₃ lie in different connected components of G₁₃, we can reverse the coloration on that containing V₁, thus assigning color number 1 to V and completing the task.

If on the contrary V₁ and V₃ lie in the same connected component of G₁₃, we can find a path in G₁₃ joining them, that is a sequence of edges and vertices painted only with colors 1 and 3.

Now turn to the subgraph G₂₄ of V' consisting of the vertices that are colored with colors 2 and 4 only, and edges connecting two of them, and apply the same arguments as before. Then either we are able to reverse a coloration on a subgraph of G₂₄ and paint V with, say, color number 2, or we can connect V₂ and V₄ with a path containing vertices colored only with colors 2 and 4. The latter possibility is clearly absurd, as such a path would intersect the path we constructed in G₁₃.

So G can in fact be five-colored, contrary to the initial presumption.

Linear time five-coloring algorithm

In 1996, Robertson, Sanders, Seymour, and Thomas described a quadratic four-coloring algorithm in their "Efficiently four-coloring planar graphs". ^[1] In the same paper they briefly describe a linear-time five-coloring algorithm, which is asymptotically optimal. In Computer science, an Algorithm is said to be asymptotically optimal if roughly speaking for large inputs it performs at worst a constant factor (independent of 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:

Wernicke's Theorem: Assume G is planar, nonempty, has no faces bounded by two edges, and has minimum degree 5. The discharging method is a technique used to prove lemmas in structural Graph theory. Then G has a vertex of degree 5 which is adjacent to a vertex of degree at most 6.

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:

• S₄: Contains all remaining vertices with either degree at most four, or degree five and at most four distinct adjacent vertices (due to multiple edges).
• S₅: Contains all remaining vertices that have degree five, five distinct adjacent vertices, and at least one adjacent vertex with degree at most six.
• S_d: Contains all vertices deleted from the graph so far, in the order that they were deleted.

The algorithm works as follows:

1. In the first step, we collapse all multiple edges to single edges, so that the graph is simple. Next, we iterate over the vertices of the graph, pushing any vertex matching the conditions for S₄ or S₅ onto the appropriate stack.
2. Next, as long as S₄ is non-empty, we pop v from S₄ and delete v from the graph, pushing it onto S_d, along with a list of its neighbors at this point in time. We check each former neighbor of v, pushing it onto S₄ or S₅ if it now meets the necessary conditions.
3. When S₄ becomes empty, we know that our graph has minimum degree five. If the graph is empty, we go to the final step 5 below. Otherwise, Wernicke's Theorem tells us that S₅ is nonempty. Pop v off S₅, delete it from the graph, and let v₁, v₂, v₃, v₄, v₅ be the former neighbors of v in clockwise planar order, where v₁ is the neighbor of degree at most 6. We check if v₁ is adjacent to v₃ (which we can do in constant time due to the degree of v₁). There are two cases:
   1. If v₁ is not adjacent to v₃, we can merge these two vertices into a single vertex. To do this, we remove v from both circular adjacency lists, and then splice the two lists together into one list at the point where v was formerly found. Provided that v maintains a reference to its position in each list, this can be done in constant time. It's possible that this might create faces bounded by two edges at the two points where the lists are spliced together; we delete one edge from any such faces.

      After doing this, we push v₃ onto S_d, along with a note that v₁ is the vertex that it was merged with. Any vertices affected by the merge are added or removed from the stacks as appropriate.

   2. Otherwise, v₂ lies inside the face outlined by v, v₁, and v₃. Consequently, v₂ cannot be adjacent to v₄, which lies outside this face. We merge v₂ and v₄ in the same manner as v₁ and v₃ above.
4. Go to step 2.
5. At this point S₄, S₅, and the graph are empty. We pop vertices off S_d. If the vertex was merged with another vertex in step 3, the vertex that it was merged with will already have been colored, and we assign it the same color. This is valid because we only merged vertices that were not adjacent in the original graph. If we had removed it in step 2 because it had at most 4 adjacent vertices, all of its neighbors at the time of its removal will have already been colored, and we can simply assign it a color that none of its neighbors is using.

See also

• Four color theorem

References

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