Four Color Theorem - the missing link
R. L. Hudson
4-05-2014
Abstract
The conjecture [1] stated that four colors are sufficient for any 2-dimensional plane map so
that no two regions with a shared border are the same color. The conjecture is now a
theorem [2], the result of a lengthy and complex proof, involving over 1000
classifications of graph objects and over 1000 hours of computer time. Instead of
considering all maps, this paper examines which elements of a map are possible and the
issue of causality.
1. Method
To simplify the analysis, map configurations will be transformed to graph objects.
fig. 1a
fig. 1b
fig. 1c
Within the regions 1 and 2 (fig.1a), a small circle (node) is formed which represents the
area of each region (fig. 1b). A line connecting the nodes represents any path crossing the
border between any two regions. The shared border must have a length greater than zero.
Removal of the original geographic borders leaves the graph representation (fig.1c). This
eliminates dealing with regions having highly irregular borders, and focuses on
connectivity. Regions cannot overlap on a real map, therefore the lines (links) between
nodes cannot cross.
1.1 The color restriction for maps can be restated for graphs as: no paired nodes are the
same color.
2. Definitions
2.1 Color number is the minimum quantity of colors required that does not violate color
restriction as stated in 1.1.
2.2 Objects are configurations of connected nodes.
2.3 Complete means all node pairs are formed.
2.4 Basic objects are complete.
2.5 A set of colors is represented by the letters {A,B,C,D,...}.
3. Proposed Cause
3.1. If node x contacts k different colored nodes in a set of n nodes with a current color
number k, then by 1.1, x requires an additional color. This means there are two
requirements to increase the color number.
3.1.1 The set contains a complete k colored object.
3.1.2 Node x has access to one of each of the k colored nodes.
Access also implies that the color number of the resulting object is determined locally
(within a one link range) and independently of the remaining n - k nodes.
4. Forming basic objects
The 1-color or 1c basic object is a single node, which in geographical terms represents an
island. This is a total state of isolation and allows the reuse of one color.
fig. 4a
fig. 4b
fig.4c
Linking nodes in one dimension (fig.4a) only requires two colors, since the added node
isolates the next additional node from all previous ones. Linking in 2 dimensions (fig.4b)
forms a closed sequence, requiring three colors (fig.4c), and replaces the 1-dimensional
isolation with 2-dimensional isolation. If fig.4c was on the surface of a sphere, it would
be obvious that interior and exterior references are irrelevant when adding nodes.
fig. 4d
fig. 4e
Adding a node to fig.4c, with maximum linking, forms fig.4d, requiring four colors. If
additional nodes are added, the 2-dimensional isolation allows reuse of one of the isolated
colors (fig.4e). The 1, 2, and 3 color basic objects satisfied condition 3.1, but the 4c
object (fig.4d) fails since it lacks access. To form a 5c object requires an additional
dimension to defeat the current state of isolation, as was done with the previous objects.
5. Completeness
Since lines cannot partition a 3D space, all pairs are possible in 3D, with total access and
no isolation. An n node object requires n colors, therefore color number k = n. The line
count L in 3D is
L3 = n(n-1)/2, with n>2.
Examining fig.4e, with maximum linking, each additional node can only form three links.
The isolated node color can be reused, and color number k becomes independent of node
count n. The line count L in 2D is
L2 = 3n-6, with n>2.
The variable m is used as a measure of completeness for the node count of a graph. It is
defined as the ratio of maximum linked pairs in 2D to maximum linked pairs in 3D.
m = L2 /L3 = 6(n-2)/n(n-1)
The value of m equals 1 for 3 and 4 nodes but is always decreasing for n >4. The
difference (1-m) may also be used as a measure of redundancy, relative to color number.
6. The causality issue and redundancy
fig. 6a
fig. 6b
Some authors use fig. 6a as an argument against completeness as the cause of color
number, reasoning that there are not 3 mutually connected nodes as in fig.4c yet it still
requires 3 colors. Fig. 6b is the 4 color counterpart for the same argument. By 3.1 the
node colored C in figure 6a is determined by the first AB pair it contacts.
fig.6c
fig.6d
If either AB pair is removed leaving a single link as in fig.6c, a third color is required.
The same procedure can be applied to fig.6b, by removing one AB pair. The color
number is determined by the complete object that remains after removal of the redundant
parts of the configuration. An alternate approach is to consider the objects before the last
node was added. In each case the new node could have been linked to a different pair
which would have been complete, making the other pairs redundant. In these cases,
redundancy obscures completeness.
7. Map Analysis
MA uses a tabular representation of a graph containing each node's location and
connective relationship to all other nodes. This enables analysis of a map without the
classification of objects by configuration. MA provides a systematic, consistent method
for assigning colors to a map, in accordance with condition 1.1. It avoids the color
conflicts which may not be apparent for large complex maps. The table has n rows and n
columns, n being the number of nodes. The map is numbered starting at an arbitrary
node, and moving to adjacent nodes, labeling them in consecutive order from 1 to n. The
columns are numbered at the top, left to right, and the number of lines leaving each node
is entered at the bottom of each column. The columns are then sorted in reverse order
with the greatest line count as column 1. The reason for this is statistically the node with
the greatest line count will have the most color restrictions. The rows are numbered the
same as the columns. The value for each cell is 1 if the row node contacts the column
node, 0 otherwise. Because contact is independent of order, the section above the
diagonal mirrors the lower and is not needed. The diagonal values are 'x' because the
values 0 and 1 only apply to pairs and they also serve as a visual aid. The colors are
assigned to each row node from top to bottom by placing a 1 in the corresponding color
column. For each row, if the cell value is 1, then a 0 is entered in the column
corresponding to the color assigned to the linked node, preventing its use for the row
node. The process is repeated for the remaining nodes, always using the first color
available from the sequence ABCD.
A
2
6
1
D
C
B
3
5
4
fig. 7a
C
D
fig. 7b
2
3
4
5
1
6
2
x
1
1
1
1
1
5
table 7.1
3 4 5 1 6
x
1 x
1 1 x
1 1 0 x
1 0 1 0 x
5 4 4 3 3
ABCD
1
01
001
0001
0001
0010
Fig.7a is the input for table 7.1, and fig.7b the result from using the coloring process in
MA.
8. Reduction
Table 1 can also be used in the reduction of a map by the removal of all redundant
components, and adjusting the table values. The color assignments in table 7.1 show
nodes 1 and 6 as repeat colors.
2
3
4
5
2
x
1
1
1
A
table 8.1
4 5 ABCD
1
01
x
001
1 x 0001
3
x
1
1
B
D
C
fig. 8a
Table 8.1 is table 7.1 reduced and fig.8a the corresponding basic 4c object.
Examining the color columns in table 7.1, the number of occurrences of ordered
sequences A B C D is 1*1*2*2 = 4. This means there are four basic 4c objects embedded
in the graph, which may be considered as a superposition of basic objects
(fig. 8a-8d).
2
2
2
1
6
3
4
fig. 8b
6
1
3
5
fig. 8c
3
fig. 8d
Any graph can be reduced to a basic object by removing the redundant elements, which
corresponds to removing the zeros from the table.
9. Causality
2
1
4
3
6
5
11
A
C
7
10
9
8
C
B
B
A
fig. 9a
C
A
C
B
12
A
B
fig.9b
6
9
2
8
10
table 9.1
1
3
4
7
5
6
9
2
8
10
1
3
4
7
11
12
5
x
1
1
1
1
0
1
0
1
0
0
0
x
1
1
0
1
0
1
0
1
0
0
x
0
1
1
0
0
0
0
1
1
x
0
0
1
1
0
0
0
0
x
0
0
0
1
0
1
0
x
0
0
0
1
0
1
x
0
1
0
0
0
x
0
1
0
0
x
0
0
0
x
0
0
x
1
6
6
6
4
4
4
3
3
3
3
3
3
11 12
x
ABCD
1
01
001
001
010
100
010
100
001
001
100
010
The isolation effect is examined using fig.9a as the input for table 9.1 with fig.9b the
resulting coloring. According to 3.1 the coloring of the 4-node sample configuration (9
thru12) is independent of nodes 1 thru 4.
5
6
9
8
10
7
11
12
5
x
1
1
1
0
0
0
0
table 9.2
6 9 8 10 7 11 12
x
1
0
1
1
0
0
x
1
1
0
1
1
x
0
0
1
0
x
1 x
0 0 x
1 0 1 x
ABCD
1
01
001
01
10
001
10
01
Table 9.2 is table 9.1 with nodes 1 thru 4 removed and produces the same coloring for the
sample. Table 9.1 has less than maximum links and the color columns show two color
restrictions for each node after row 2. This means there are two remaining colors to
choose. Fewer links results in less color restriction and defeats any attempt to increase
color number.
10. Conclusion
Using maximum links, the color number for a 2D plane is determined with the first four
nodes, as shown in fig.4d. Additional nodes only generate more node triads, isolation
allows the reuse of one color, and the color number becomes independent of the node
count. Once color number is determined, the remainder of the map is redundant as shown
in section 5.
If less than maximum links are formed, and the plane is uniformly covered with
triangulated nodes (boundaries) as in fig.9b, with a color number of 3, a new node added
with maximum links to any randomly selected boundary would require a fourth color.
This supports statement 3.1. When the map is divided into boundaries, the coloring
process operates in each boundary independently of the others. The notion of causality
spreading through the links, and therefore a larger map providing a possible configuration
for the 5c object, has been shown to be a false assumption. If all the links of a map were
necessary, none could be removed, and reduction would not be possible. As shown in
section 8, for k=1 to 4, a k-colored map can be reduced to a basic k-colored object.
1
2
3
4
2D 5
table 10.1
1 2 3 4 5
x
1 x
1 1 x
1 1 1 x
1 1 1 0 x
3D 5 1 1 1
1 x
ABCDE
1
01
001
0001
0001
00001
Table 10.1 represents the 4c object (fig.4e) in 2D with the last row in 3D for comparison.
Node 5 in 2D is isolated from node 4 allowing color D to be reused. Node 5 in 3D is
linked to node 4 requiring a fifth color E. The difference is the extra dimension defeating
the isolation in 2D. The 5c object (fig.10a) requires an additional dimension for linking,
and therefore a torus for its representation as neighboring regions (fig.10b).
fig.10a
fig.10b
references
[1] http://en.wikipedia.org/wiki/Four color theorem
[2] Appel, Kenneth; Haken, Wolfgang; Koch, John (1977), "Every Planar Map is Four
Colorable", Illinois Journal of Mathematics 21: 439–567