user
A Graph is a collection of nodes
(or vertices), connected by edges (or not).
Nodes directly connected by edges are called neighbours.

In our representation of graphs, nodes are numbered and edges are represented
by the two node numbers connected by the edge separated by a dash.
Edges define the nodes being connected. Only unconnected nodes *need* a separate
description.

For example,

```
0-1 1-2 2-0 3
```

Describes the following graph. Note that node 3 has no neighbours

Example graph

```
+---+
| 3 |
+---+

  +-------------------+
  |                   |
+---+     +---+     +---+
| 0 | --- | 1 | --- | 2 |
+---+     +---+     +---+

```

A useful internal datastructure for a graph and for later graph algorithms is
as a mapping between each node and the set/list of its neighbours.

In the above example:

```
0 maps-to 1 and 2
1 maps to 2 and 0
2 maps-to 1 and 0
3 maps-to <nothing>
```

Graph colouring task

Colour the vertices of a given graph so that no edge is between verticies of
the same colour.

* Integers may be used to denote different colours.
* Algorithm should do better than just assigning each vertex a separate colour. The idea is to minimise the number of colours used, although no algorithm short of exhaustive search for the minimum is known at present, (and exhaustive search is **not** a requirement).
* Show for each edge, the colours assigned on each vertex.
* Show the total number of nodes, edges, and colours used for each graph.

Use the following graphs


Ex1

```
       0-1 1-2 2-0 3

```

```
+---+
| 3 |
+---+

  +-------------------+
  |                   |
+---+     +---+     +---+
| 0 | --- | 1 | --- | 2 |
+---+     +---+     +---+

```

Ex2

The wp articles left-side graph

```
   1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7

```

```


  +----------------------------------+
  |                                  |
  |                      +---+       |
  |    +-----------------| 3 | ------+----+
  |    |                 +---+       |    |
  |    |                   |         |    |
  |    |                   |         |    |
  |    |                   |         |    |
  |  +---+     +---+     +---+     +---+  |
  |  | 8 | --- | 1 | --- | 6 | --- | 4 |  |
  |  +---+     +---+     +---+     +---+  |
  |    |         |                   |    |
  |    |         |                   |    |
  |    |         |                   |    |
  |    |       +---+     +---+     +---+  |
  +----+------ | 7 | --- | 2 | --- | 5 | -+
       |       +---+     +---+     +---+
       |                   |
       +-------------------+

```

Ex3

The wp articles right-side graph which is the same graph as Ex2, but with
different node orderings and namings.

```
   1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6

```

```


  +----------------------------------+
  |                                  |
  |                      +---+       |
  |    +-----------------| 5 | ------+----+
  |    |                 +---+       |    |
  |    |                   |         |    |
  |    |                   |         |    |
  |    |                   |         |    |
  |  +---+     +---+     +---+     +---+  |
  |  | 8 | --- | 1 | --- | 4 | --- | 7 |  |
  |  +---+     +---+     +---+     +---+  |
  |    |         |                   |    |
  |    |         |                   |    |
  |    |         |                   |    |
  |    |       +---+     +---+     +---+  |
  +----+------ | 6 | --- | 3 | --- | 2 | -+
       |       +---+     +---+     +---+
       |                   |
       +-------------------+

```

Ex4

This is the same graph, node naming, and edge order as Ex2 except some of the edges x-y are flipped to y-x.
This might alter the node order used in the greedy algorithm leading to differing numbers of colours.

```
   1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7

```

```


                      +-------------------------------------------------+
                      |                                                 |
                      |                                                 |
  +-------------------+---------+                                       |
  |                   |         |                                       |
+---+     +---+     +---+     +---+     +---+     +---+     +---+     +---+
| 4 | --- | 5 | --- | 2 | --- | 7 | --- | 1 | --- | 6 | --- | 3 | --- | 8 |
+---+     +---+     +---+     +---+     +---+     +---+     +---+     +---+
  |         |                             |         |         |         |
  +---------+-----------------------------+---------+         |         |
            |                             |                   |         |
            |                             |                   |         |
            +-----------------------------+-------------------+         |
                                          |                             |
                                          |                             |
                                          +-----------------------------+

```

References

* Greedy coloring Wikipedia.
* Graph Coloring : Greedy Algorithm & Welsh Powell Algorithm by Priyank Jain.
assistant
(formerly Perl 6)

```
sub GraphNodeColor(@RAW) {
   my %OneMany = my %NodeColor;
   for @RAW { %OneMany{$_[0]}.push: $_[1] ; %OneMany{$_[1]}.push: $_[0] }
   my @ColorPool = "0", "1" … ^+%OneMany.elems; # as string
   my %NodePool  = %OneMany.BagHash; # this DWIM is nice
   if %OneMany<NaN>:exists { %NodePool{$_}:delete for %OneMany<NaN>, NaN } # pending
   while %NodePool.Bool {
      my $color = @ColorPool.shift;
      my %TempPool = %NodePool;
      while (my \n = %TempPool.keys.sort.first) {
         %NodeColor{n} = $color;
         %TempPool{n}:delete;
         %TempPool{$_}:delete for @(%OneMany{n}) ; # skip neighbors as well
         %NodePool{n}:delete;
      }
   }
   if %OneMany<NaN>:exists { # islanders use an existing color
      %NodeColor{$_} = %NodeColor.values.sort.first for @(%OneMany<NaN>)
   }
   return %NodeColor
}

my \DATA = [
   [<0 1>,<1 2>,<2 0>,<3 NaN>,<4 NaN>,<5 NaN>],
   [<1 6>,<1 7>,<1 8>,<2 5>,<2 7>,<2 8>,<3 5>,<3 6>,<3 8>,<4 5>,<4 6>,<4 7>],
   [<1 4>,<1 6>,<1 8>,<3 2>,<3 6>,<3 8>,<5 2>,<5 4>,<5 8>,<7 2>,<7 4>,<7 6>],
   [<1 6>,<7 1>,<8 1>,<5 2>,<2 7>,<2 8>,<3 5>,<6 3>,<3 8>,<4 5>,<4 6>,<4 7>],
];

for DATA {
   say "DATA   : ", $_;
   say "Result : ";
   my %out = GraphNodeColor $_;
   say "$_[0]-$_[1]:\t Color %out{$_[0]} ",$_[1].isNaN??''!!%out{$_[1]} for @$_;
   say "Nodes  : ", %out.keys.elems;
   say "Edges  : ", $_.elems;
   say "Colors : ", %out.values.Set.elems;
}

```

Output:

```
DATA   : [(0 1) (1 2) (2 0) (3 NaN) (4 NaN) (5 NaN)]
Result :
0-1:     Color 0 1
1-2:     Color 1 2
2-0:     Color 2 0
3-NaN:   Color 0
4-NaN:   Color 0
5-NaN:   Color 0
Nodes  : 6
Edges  : 6
Colors : 3
DATA   : [(1 6) (1 7) (1 8) (2 5) (2 7) (2 8) (3 5) (3 6) (3 8) (4 5) (4 6) (4 7)]
Result :
1-6:     Color 0 1
1-7:     Color 0 1
1-8:     Color 0 1
2-5:     Color 0 1
2-7:     Color 0 1
2-8:     Color 0 1
3-5:     Color 0 1
3-6:     Color 0 1
3-8:     Color 0 1
4-5:     Color 0 1
4-6:     Color 0 1
4-7:     Color 0 1
Nodes  : 8
Edges  : 12
Colors : 2
DATA   : [(1 4) (1 6) (1 8) (3 2) (3 6) (3 8) (5 2) (5 4) (5 8) (7 2) (7 4) (7 6)]
Result :
1-4:     Color 0 1
1-6:     Color 0 2
1-8:     Color 0 3
3-2:     Color 1 0
3-6:     Color 1 2
3-8:     Color 1 3
5-2:     Color 2 0
5-4:     Color 2 1
5-8:     Color 2 3
7-2:     Color 3 0
7-4:     Color 3 1
7-6:     Color 3 2
Nodes  : 8
Edges  : 12
Colors : 4
DATA   : [(1 6) (7 1) (8 1) (5 2) (2 7) (2 8) (3 5) (6 3) (3 8) (4 5) (4 6) (4 7)]
Result :
1-6:     Color 0 1
7-1:     Color 1 0
8-1:     Color 1 0
5-2:     Color 1 0
2-7:     Color 0 1
2-8:     Color 0 1
3-5:     Color 0 1
6-3:     Color 1 0
3-8:     Color 0 1
4-5:     Color 0 1
4-6:     Color 0 1
4-7:     Color 0 1
Nodes  : 8
Edges  : 12
Colors : 2
```