Five color theorem
Encyclopedia
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.
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. The four color theorem was finally proven by Kenneth Appel
and Wolfgang Haken
at the University of Illinois, with the aid of a computer
. They were assisted in some algorithmic work by John A. Koch.
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.
The proof relies on the Euler characteristic
to show that there must be a vertex
shared by at most five edges, and on the fact that 
is planar
, i.e. it may be embedded in the plane without intersecting edges.
Now remove
from 
. The graph 
obtained this way has one fewer vertex than 
, so we can assume by induction
that it can be colored with only five colors.
must be connected to five other vertices, since if not it can be colored in 
with a color not used by them. So now look at those five vertices 
, 
, 
, 
, 
that were adjacent to 
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 
in a consistent way to render our graph 5-colored.
So we can assume that
, 
, 
, 
, 
are colored with colors 1, 2, 3, 4, 5 respectively.
Now consider the subgraph
of 
consisting of the vertices that are colored with colors 1 and 3 only, and edges connecting two of them. If 
and 
lie in different connected components of 
, we can reverse the coloration on that containing 
, thus assigning color number 1 to 
and completing the task.
If on the contrary
and 
lie in the same connected component of 
, we can find a path in 
joining them, that is a sequence of edges and vertices painted only with colors 1 and 3.
Now turn to the subgraph
of 
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 
and paint 
with, say, color number 2, or we can connect 
and 
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 
.
So
can in fact be five-colored, contrary to the initial presumption.
. 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:
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
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.
The five color theorem is implied by the stronger four color theorem
Four color theorem
In mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color...
, but is considerably easier to prove. It was based on a failed attempt at the four color proof by Alfred Kempe
Alfred Kempe
Sir Alfred Bray Kempe D.C.L. F.R.S. was a mathematician best known for his work on linkages and the four color theorem....
in 1879. Percy John Heawood
Percy John Heawood
Percy John Heawood was a British mathematician educated at Queen Elizabeth's School, Ipswich, and Exeter College, Oxford....
found an error 11 years later, and proved the five color theorem based on Kempe's work. The four color theorem was finally proven by Kenneth Appel
Kenneth Appel
Kenneth Ira Appel is a mathematician who in 1976, with colleague Wolfgang Haken at the University of Illinois at Urbana-Champaign, solved one of the most famous problems in mathematics, the four-color theorem...
and Wolfgang Haken
Wolfgang Haken
Wolfgang Haken is a mathematician who specializes in topology, in particular 3-manifolds.In 1976 together with colleague Kenneth Appel at the University of Illinois at Urbana-Champaign, Haken solved one of the most famous problems in mathematics, the four-color theorem...
at the University of Illinois, with the aid of a computer
Computer-assisted proof
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer.Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and...
. They were assisted in some algorithmic work by John A. Koch.
Outline of the proof by contradiction
First of all, one associates a planar graphGraph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
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.The proof relies on the Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
to show that there must be a vertex
shared by at most five edges, and on the fact that is planarPlanar graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints...
, i.e. it may be embedded in the plane without intersecting edges.
Now remove
from . The graph obtained this way has one fewer vertex than , so we can assume by inductionMathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
that it can be colored with only five colors.
must be connected to five other vertices, since if not it can be colored in with a color not used by them. So now look at those five vertices , , , , that were adjacent to 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 in a consistent way to render our graph 5-colored.So we can assume that
, , , , are colored with colors 1, 2, 3, 4, 5 respectively.Now consider the subgraph
of consisting of the vertices that are colored with colors 1 and 3 only, and edges connecting two of them. If and lie in different connected components of , we can reverse the coloration on that containing , thus assigning color number 1 to and completing the task.If on the contrary
and lie in the same connected component of , we can find a path in joining them, that is a sequence of edges and vertices painted only with colors 1 and 3.Now turn to the subgraph
of 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 and paint with, say, color number 2, or we can connect and 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 .So
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". In the same paper they briefly describe a linear-time five-coloring algorithm, which is asymptotically optimalAsymptotically 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 worse than the best possible algorithm...
. 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 degreeDegree (graph theory)In graph theory, the degree of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice. The degree of a vertex v is denoted \deg. The maximum degree of a graph G, denoted by Δ, and the minimum degree of a graph, denoted by δ, are the maximum and minimum degree...
5. 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.
- Sd: Contains all vertices deleted from the graph so far, in the order that they were deleted.
The algorithm works as follows:
- 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.
- Next, as long as S4 is non-empty, we pop v from S4 and delete v from the graph, pushing it onto Sd, 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.
- 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:
- 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 Sd, 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.
- 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.
- 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.
- Go to step 2.
- At this point S4, S5, and the graph are empty. We pop vertices off Sd. 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.