## Coloring The Edges of an Icosahedron

```It's possible to color the edges of an icosahedron in five different
colors such that the colors of the edges meeting at each vertex are
all different.  In fact, this can be done in several distinct ways.
The exact number of distinct ways depends on how we define "distinct-
ness".  If we fix the orientation of the icosahedron, and assign the
five colors a,b,c,d,e to the five edges that meet at the "top" vertex,
then there are 780 distinct ways of coloring the rest of the edges
such that each color adjoins each vertex.  However, if rotation of
the icosahedron is allowed, many of these colorings can be shown to
be equivalent.

One way of defining the "fundamentally distinct" colorings is to
classify them according to the numbers of distinct vertices.  We
can characterize each vertex by the cycle of colors that meet at
that point, taken in clockwise order.  Two vertices are said to
be equivalent if they have the same cycle of colors.  We could then
say that two colorings are equivalent if they have the same numbers
of distinct vertices.  On this basis there are just seven distinct
colorings, with the valencies shown below:

coloring     1   2   3   4   5      number
--------    --- --- --- --- ---   ---------
#1        5   2   1   0   0       120
#2        6   3   0   0   0       120
#3        0   6   0   0   0       150
#4        8   2   0   0   0       120
#5        7   0   0   0   1        24
#6       12   0   0   0   0       216
#7        0   4   0   1   0        30
--------
total         780

For example, each of the 120 colorings in the 1st set contains 5
unique vertices, 2 sets of two equivalent vertices, and 1 set of
three equivalent vertices.

This shows that there are at least seven truely distinct colorings,
because obviously no rotations of the icosahedron or permutations of
the colors will change the vertex valencies listed above.  However,
it's possible that within some (or all) of these seven categories
there are distinct colorings in the sense that, even though they
have the same valencies, they cannot be transformed into each other
by rotations or permutations.

In the preceeding discussion I split the 780 colorings into seven
classes, according to the multiplicities of their vertex equivalence
classes.  (Two vertices are regarded as equivalent if they have the
same cycle of colors.)  Now I've split each class into sub-classes.
Each coloring in a given sub-class has the same twelve vertex
cycles, up to permutations of the colors.

vertex
multiplicities
class  1  2  3  4  5   total         sub-classes
-----  -  -  -  -  -   -----   --------------------------------
#1    5  2  1  0  0    120    60 60
#2    6  3  0  0  0    120    60 60
#3    0  6  0  0  0    150    30 30 30 30 15 15
#4    8  2  0  0  0    120    60 60
#5    7  0  0  0  1     24    24
#6   12  0  0  0  0    216    30 30 30 30 30 30 20 10 6
#7    0  4  0  1  0     30    30
-----
total        780

This shows that there are 23 sub-classes, so there are at least 23
truly distinct colorings that cannot be transformed into each other
by rotations of the icosahedron and permutations of the colors.

By checking every orientation of the icosahedron and every permutation
of the colors it turns out that the 23 sub-classes can be further
divided into 29 truly distinct colorings.  The populations of these
29 colorings are shown below:

vertex
multiplicities           breakdown of subclasses
class  1  2  3  4  5   total    into distinct colorings
-----  -  -  -  -  -   -----   --------------------------------
#1    5  2  1  0  0    120     60 60
#2    6  3  0  0  0    120     60 60
#3    0  6  0  0  0    150     30 30  (15 15) (15 15) 15 15
#4    8  2  0  0  0    120     60 60
#5    7  0  0  0  1     24    (12 12)
#6   12  0  0  0  0    216     30 30 30 30 30 (15 15) (10 10) 10 (5 1)
#7    0  4  0  1  0     30     30
-----
total        780

Incidentally, the coloring in class #6 with just one representative
out of the total 780 is the unique pattern such that all five colors
have an identical arrangement (up to rotation), and not only does
each color adjoin each vertex only once, but each color occurs only
once on the perimeter of the pentagon surrounding each vertex.  With
this pattern, the pentagon edge with a given color is always opposite
the "spoke" with that color.  It follows that, since no two vertices
have the same color cycle, no two pentagon perimeters have the same
color cycle.

Here's a summary of my classes.  Each coloring is represented by a
string of 30 digits in the range 1 to 5, signifying which of the five
colors is applied to the respective edge.  The order of the edges in
the strings is as follows (in terms of the vertices numbered 1 - 12)

1 [1,2]    7 [2,6]    13 [4,5]    19 [6,11]    25 [7,12]
2 [1,3]    8 [2,11]   14 [4,8]    20 [6,10]    26 [8,9]
3 [1,4]    9 [2,7]    15 [4,9]    21 [11,7]    27 [8,12]
4 [1,5]   10 [3,4]    16 [5,6]    22 [11,10]   28 [9,10]
5 [1,6]   11 [3,7]    17 [5,9]    23 [11,12]   29 [9,12]
6 [2,3]   12 [3,8]    18 [5,10]   24 [7,8]     30 [10,12]

Icosahedron coloring pattern     size of class

1       123453245145245135341253221143       60
2       123453245145245135342153112243       60
3       123453245145245315142351332241       60
4       123453245145245315143251223341       60
5       123453245145524123342151334512       30
6       123453245145524123342153114532       60
7       123453245145524123342511334152       15
8       123453245145524321142353114532       15
9       123453245145542312142351332541       12
10       123453245415125325142353441132       60
11       123453245415125325143254231143       12
12       123453245415215315142353442231       30
13       123453245415215315143252443321       15
14       123453245415512132342513442153       30
15       123453245415521321142353441532       30
16       123453245415521321142533441352       10
17       123453245514124325142353451132       30
18       123453245514124352142533415321       30
19       123453245514214315142353452231       10
20       123453245514214351142533425312        5
21       123453245541124325142353154132       15
22       123453245541142352143522135143       30
23       123453245541241135342153152243       30
24       123453245541241153341522335241       15
25       123453245541241351143522135243       15
26       123453425145542213311542331452       15
27       123453425541142235133452153142       15
28       123453425541142253311542335412       10
29       123454325531142253411532435412        1

As Henk Penning pointed out, the size of each of these equivalence
classes must divide 60, per the orbit/stabilizer theorem.  Here's an
illustration of the 29 distinct edge-colorings of the icosahedron
with five colors such that each color touches each vertex:

```