# Graph colouring

Graph colouring
You are encouraged to solve this task according to the task description, using any language you may know.

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>```

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

## Go

As mentioned in the task description, there is no known efficient algorithm which can guarantee that a minimum number of colors is used for a given graph. The following uses both the so-called 'greedy' algorithm (as described here) and the Welsh-Powell algorithm (as described here), suitably adjusted to the needs of this task.

The results are exactly the same for both algorithms. Whilst one would normally expect Welsh-Powell to give better results overall, the last three examples are not well suited to it as each node has exactly the same number of neighbors i.e. the valences are equal.

The results agree with the Python entry for examples 1 and 2 but, for example 3, Python gives 2 colors compared to my 4 and, for example 4, Python gives 3 colors compared to my 2.

`package main import (    "fmt"    "sort") type graph struct {    nn  int     // number of nodes    st  int     // node numbering starts from    nbr [][]int // neighbor list for each node} type nodeval struct {    n int // number of node    v int // valence of node i.e. number of neighbors} func contains(s []int, n int) bool {    for _, e := range s {        if e == n {            return true        }    }    return false} func newGraph(nn, st int) graph {    nbr := make([][]int, nn)    return graph{nn, st, nbr}} // Note that this creates a single 'virtual' edge for an isolated node.func (g graph) addEdge(n1, n2 int) {    n1, n2 = n1-g.st, n2-g.st // adjust to starting node number    g.nbr[n1] = append(g.nbr[n1], n2)    if n1 != n2 {        g.nbr[n2] = append(g.nbr[n2], n1)    }} // Uses 'greedy' algorithm.func (g graph) greedyColoring() []int {    // create a slice with a color for each node, starting with color 0    cols := make([]int, g.nn) // all zero by default including the first node    for i := 1; i < g.nn; i++ {        cols[i] = -1 // mark all nodes after the first as having no color assigned (-1)    }    // create a bool slice to keep track of which colors are available    available := make([]bool, g.nn) // all false by default    // assign colors to all nodes after the first    for i := 1; i < g.nn; i++ {        // iterate through neighbors and mark their colors as available        for _, j := range g.nbr[i] {            if cols[j] != -1 {                available[cols[j]] = true            }        }        // find the first available color        c := 0        for ; c < g.nn; c++ {            if !available[c] {                break            }        }        cols[i] = c // assign it to the current node        // reset the neighbors' colors to unavailable        // before the next iteration        for _, j := range g.nbr[i] {            if cols[j] != -1 {                available[cols[j]] = false            }        }    }    return cols} // Uses Welsh-Powell algorithm.func (g graph) wpColoring() []int {    // create nodeval for each node    nvs := make([]nodeval, g.nn)    for i := 0; i < g.nn; i++ {        v := len(g.nbr[i])        if v == 1 && g.nbr[i][0] == i { // isolated node            v = 0        }        nvs[i] = nodeval{i, v}    }    // sort the nodevals in descending order by valence    sort.Slice(nvs, func(i, j int) bool {        return nvs[i].v > nvs[j].v    })    // create colors slice with entries for each node    cols := make([]int, g.nn)    for i := range cols {        cols[i] = -1 // set all nodes to no color (-1) initially    }    currCol := 0 // start with color 0    for f := 0; f < g.nn-1; f++ {        h := nvs[f].n        if cols[h] != -1 { // already assigned a color            continue        }        cols[h] = currCol        // assign same color to all subsequent uncolored nodes which are        // not connected to a previous colored one    outer:        for i := f + 1; i < g.nn; i++ {            j := nvs[i].n            if cols[j] != -1 { // already colored                continue            }            for k := f; k < i; k++ {                l := nvs[k].n                if cols[l] == -1 { // not yet colored                    continue                }                if contains(g.nbr[j], l) {                    continue outer // node j is connected to an earlier colored node                }            }            cols[j] = currCol        }        currCol++    }    return cols} func main() {    fns := [](func(graph) []int){graph.greedyColoring, graph.wpColoring}    titles := []string{"'Greedy'", "Welsh-Powell"}    nns := []int{4, 8, 8, 8}    starts := []int{0, 1, 1, 1}    edges1 := [][2]int{{0, 1}, {1, 2}, {2, 0}, {3, 3}}    edges2 := [][2]int{{1, 6}, {1, 7}, {1, 8}, {2, 5}, {2, 7}, {2, 8},        {3, 5}, {3, 6}, {3, 8}, {4, 5}, {4, 6}, {4, 7}}    edges3 := [][2]int{{1, 4}, {1, 6}, {1, 8}, {3, 2}, {3, 6}, {3, 8},        {5, 2}, {5, 4}, {5, 8}, {7, 2}, {7, 4}, {7, 6}}    edges4 := [][2]int{{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 j, fn := range fns {        fmt.Println("Using the", titles[j], "algorithm:\n")        for i, edges := range [][][2]int{edges1, edges2, edges3, edges4} {            fmt.Println("  Example", i+1)            g := newGraph(nns[i], starts[i])            for _, e := range edges {                g.addEdge(e[0], e[1])            }            cols := fn(g)            ecount := 0 // counts edges            for _, e := range edges {                if e[0] != e[1] {                    fmt.Printf("    Edge  %d-%d -> Color %d, %d\n", e[0], e[1],                        cols[e[0]-g.st], cols[e[1]-g.st])                    ecount++                } else {                    fmt.Printf("    Node  %d   -> Color %d\n", e[0], cols[e[0]-g.st])                }            }            maxCol := 0 // maximum color number used            for _, col := range cols {                if col > maxCol {                    maxCol = col                }            }            fmt.Println("    Number of nodes  :", nns[i])            fmt.Println("    Number of edges  :", ecount)            fmt.Println("    Number of colors :", maxCol+1)            fmt.Println()        }    }}`
Output:
```Using the 'Greedy' algorithm:

Example 1
Edge  0-1 -> Color 0, 1
Edge  1-2 -> Color 1, 2
Edge  2-0 -> Color 2, 0
Node  3   -> Color 0
Number of nodes  : 4
Number of edges  : 3
Number of colors : 3

Example 2
Edge  1-6 -> Color 0, 1
Edge  1-7 -> Color 0, 1
Edge  1-8 -> Color 0, 1
Edge  2-5 -> Color 0, 1
Edge  2-7 -> Color 0, 1
Edge  2-8 -> Color 0, 1
Edge  3-5 -> Color 0, 1
Edge  3-6 -> Color 0, 1
Edge  3-8 -> Color 0, 1
Edge  4-5 -> Color 0, 1
Edge  4-6 -> Color 0, 1
Edge  4-7 -> Color 0, 1
Number of nodes  : 8
Number of edges  : 12
Number of colors : 2

Example 3
Edge  1-4 -> Color 0, 1
Edge  1-6 -> Color 0, 2
Edge  1-8 -> Color 0, 3
Edge  3-2 -> Color 1, 0
Edge  3-6 -> Color 1, 2
Edge  3-8 -> Color 1, 3
Edge  5-2 -> Color 2, 0
Edge  5-4 -> Color 2, 1
Edge  5-8 -> Color 2, 3
Edge  7-2 -> Color 3, 0
Edge  7-4 -> Color 3, 1
Edge  7-6 -> Color 3, 2
Number of nodes  : 8
Number of edges  : 12
Number of colors : 4

Example 4
Edge  1-6 -> Color 0, 1
Edge  7-1 -> Color 1, 0
Edge  8-1 -> Color 1, 0
Edge  5-2 -> Color 1, 0
Edge  2-7 -> Color 0, 1
Edge  2-8 -> Color 0, 1
Edge  3-5 -> Color 0, 1
Edge  6-3 -> Color 1, 0
Edge  3-8 -> Color 0, 1
Edge  4-5 -> Color 0, 1
Edge  4-6 -> Color 0, 1
Edge  4-7 -> Color 0, 1
Number of nodes  : 8
Number of edges  : 12
Number of colors : 2

Using the Welsh-Powell algorithm:

Example 1
Edge  0-1 -> Color 0, 1
Edge  1-2 -> Color 1, 2
Edge  2-0 -> Color 2, 0
Node  3   -> Color 0
Number of nodes  : 4
Number of edges  : 3
Number of colors : 3

Example 2
Edge  1-6 -> Color 0, 1
Edge  1-7 -> Color 0, 1
Edge  1-8 -> Color 0, 1
Edge  2-5 -> Color 0, 1
Edge  2-7 -> Color 0, 1
Edge  2-8 -> Color 0, 1
Edge  3-5 -> Color 0, 1
Edge  3-6 -> Color 0, 1
Edge  3-8 -> Color 0, 1
Edge  4-5 -> Color 0, 1
Edge  4-6 -> Color 0, 1
Edge  4-7 -> Color 0, 1
Number of nodes  : 8
Number of edges  : 12
Number of colors : 2

Example 3
Edge  1-4 -> Color 0, 1
Edge  1-6 -> Color 0, 2
Edge  1-8 -> Color 0, 3
Edge  3-2 -> Color 1, 0
Edge  3-6 -> Color 1, 2
Edge  3-8 -> Color 1, 3
Edge  5-2 -> Color 2, 0
Edge  5-4 -> Color 2, 1
Edge  5-8 -> Color 2, 3
Edge  7-2 -> Color 3, 0
Edge  7-4 -> Color 3, 1
Edge  7-6 -> Color 3, 2
Number of nodes  : 8
Number of edges  : 12
Number of colors : 4

Example 4
Edge  1-6 -> Color 0, 1
Edge  7-1 -> Color 1, 0
Edge  8-1 -> Color 1, 0
Edge  5-2 -> Color 1, 0
Edge  2-7 -> Color 0, 1
Edge  2-8 -> Color 0, 1
Edge  3-5 -> Color 0, 1
Edge  6-3 -> Color 1, 0
Edge  3-8 -> Color 0, 1
Edge  4-5 -> Color 0, 1
Edge  4-6 -> Color 0, 1
Edge  4-7 -> Color 0, 1
Number of nodes  : 8
Number of edges  : 12
Number of colors : 2
```

## Julia

Uses a repeated randomization of node color ordering to seek a minimum number of colors needed.

`using Random """Useful constants for the colors to be selected for nodes of the graph"""const colors4 = ["blue", "red", "green", "yellow"]const badcolor = "black"@assert(!(badcolor in colors4)) """    struct graph undirected simple graphconstructed from its name and a string listing of point to point connections"""mutable struct Graph    name::String    g::Dict{Int, Vector{Int}}    nodecolor::Dict{Int, String}    function Graph(nam::String, s::String)        gdic = Dict{Int, Vector{Int}}()        for p in eachmatch(r"(\d+)-(\d+)|(\d+)(?!\s*-)" , s)            if p != nothing                if p[3] != nothing                    n3 = parse(Int, p[3])                    get!(gdic, n3, [])                else                    n1, n2 = parse(Int, p[1]), parse(Int, p[2])                    p1vec = get!(gdic, n1, [])                    !(n2 in p1vec) && push!(p1vec, n2)                    p2vec = get!(gdic, n2, [])                    !(n1 in p2vec) && push!(p2vec, n1)                end            end        end        new(nam, gdic, Dict{Int, String}())    endend """    tryNcolors!(gr::Graph, N, maxtrials) Try up to maxtrials to get a coloring with <= N colors"""function tryNcolors!(gr::Graph, N, maxtrials)    t, mintrial, minord = N, N + 1, Dict()    for _ in 1:maxtrials        empty!(gr.nodecolor)        ordering = shuffle(collect(keys(gr.g)))        for node in ordering            usedneighborcolors = [gr.nodecolor[c] for c in gr.g[node] if haskey(gr.nodecolor, c)]            gr.nodecolor[node] = badcolor            for c in colors4[1:N]                if !(c in usedneighborcolors)                    gr.nodecolor[node] = c                    break                end            end        end        t = length(unique(values(gr.nodecolor)))        if t < mintrial            mintrial = t            minord = deepcopy(gr.nodecolor)        end    end    if length(minord) > 0        gr.nodecolor = minord    endend  """    prettyprintcolors(gr::graph) print out the colored nodes in graph"""function prettyprintcolors(gr::Graph)    println("\nColors for the graph named ", gr.name, ":")    edgesdone = Vector{Vector{Int}}()    for (node, neighbors) in gr.g        if !isempty(neighbors)            for n in neighbors                edge = node < n ? [node, n] : [n, node]                if !(edge in edgesdone)                    println("    ", edge[1], "-", edge[2], " Color: ",                        gr.nodecolor[edge[1]], ", ", gr.nodecolor[edge[2]])                    push!(edgesdone, edge)                end            end        else            println("    ", node, ": ", gr.nodecolor[node])        end    end    println("\n", length(unique(keys(gr.nodecolor))), " nodes, ",        length(edgesdone), " edges, ",        length(unique(values(gr.nodecolor))), " colors.")end for (name, txt) in [("Ex1", "0-1 1-2 2-0 3"),    ("Ex2", "1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7"),    ("Ex3", "1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6"),    ("Ex4", "1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7")]    exgraph = Graph(name, txt)    tryNcolors!(exgraph, 4, 100)    prettyprintcolors(exgraph)end `
Output:
```Colors for the graph named Ex1:
0-1 Color: red, blue
0-2 Color: red, green
1-2 Color: blue, green
3: blue

4 nodes, 3 edges, 3 colors.

Colors for the graph named Ex2:
1-7 Color: blue, red
2-7 Color: blue, red
4-7 Color: blue, red
4-5 Color: blue, red
4-6 Color: blue, red
2-5 Color: blue, red
2-8 Color: blue, red
3-5 Color: blue, red
3-6 Color: blue, red
3-8 Color: blue, red
1-8 Color: blue, red
1-6 Color: blue, red

8 nodes, 12 edges, 2 colors.

Colors for the graph named Ex3:
2-7 Color: red, blue
4-7 Color: red, blue
6-7 Color: red, blue
1-4 Color: blue, red
4-5 Color: red, blue
2-3 Color: red, blue
2-5 Color: red, blue
3-6 Color: blue, red
3-8 Color: blue, red
1-8 Color: blue, red
5-8 Color: blue, red
1-6 Color: blue, red

8 nodes, 12 edges, 2 colors.

Colors for the graph named Ex4:
1-7 Color: blue, red
2-7 Color: blue, red
4-7 Color: blue, red
4-5 Color: blue, red
4-6 Color: blue, red
2-5 Color: blue, red
2-8 Color: blue, red
3-5 Color: blue, red
3-6 Color: blue, red
3-8 Color: blue, red
1-8 Color: blue, red
1-6 Color: blue, red

8 nodes, 12 edges, 2 colors.
```

## Phix

Exhaustive search, trims search space to < best so far, newused improves on unique().
Many more examples/testing would be needed before I would trust this the tiniest bit.
NB: As per talk page, when writing this I did not remotely imagine it might be used on over 400,000 nodes with over 3 million links...

`-- demo\rosetta\Graph_colouring.exwconstant tests = split("""0-1 1-2 2-0 31-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-71-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-61-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7""","\n",true) function colour(sequence nodes, links, colours, soln, integer best, next, used=0)-- fill/try each colours[next], recursing as rqd and saving any improvements.-- nodes/links are read-only here, colours is the main workspace, soln/best are-- the results, next is 1..length(nodes), and used is length(unique(colours)).-- On really big graphs I might consider making nodes..best static, esp colours,-- in which case you will probably also want a "colours[next] = 0" reset below.    integer c = 1    while c<best do        bool avail = true        for i=1 to length(links[next]) do            if colours[links[next][i]]==c then                avail = false                exit            end if        end for        if avail then            colours[next] = c            integer newused = used + (find(c,colours)==next)            if next<length(nodes) then                {best,soln} = colour(nodes,links,colours,soln,best,next+1,newused)            elsif newused<best then                {best,soln} = {newused,colours}            end if        end if        c += 1    end while    return {best,soln}end function     function add_node(sequence nodes, links, string n)    integer rdx = find(n,nodes)    if rdx=0 then        nodes = append(nodes,n)        links = append(links,{})        rdx = length(nodes)    end if    return {nodes, links, rdx}end function for t=1 to length(tests) do    string tt = tests[t]    sequence lt = split(tt," "),             nodes = {},             links = {}    integer linkcount = 0, left, right    for l=1 to length(lt) do        sequence ll = split(lt[l],"-")        {nodes, links, left} = add_node(nodes,links,ll[1])        if length(ll)=2 then            {nodes, links, right} = add_node(nodes,links,ll[2])            links[left] &= right            links[right] &= left            linkcount += 1        end if    end for    integer ln = length(nodes)    printf(1,"test%d: %d nodes, %d edges, ",{t,ln,linkcount})    sequence colours = repeat(0,ln),             soln = tagset(ln) -- fallback solution    integer next = 1, best = ln    printf(1,"%d colours:%v\n",colour(nodes,links,colours,soln,best,next))end for`
Output:
```test1: 4 nodes, 3 edges, 3 colours:{1,2,3,1}
test2: 8 nodes, 12 edges, 2 colours:{1,2,2,2,1,2,1,1}
test3: 8 nodes, 12 edges, 2 colours:{1,2,2,2,1,2,1,1}
test4: 8 nodes, 12 edges, 2 colours:{1,2,2,2,2,1,1,1}
```

## Python

`import refrom collections import defaultdictfrom itertools import count  connection_re = r"""    (?: (?P<N1>\d+) - (?P<N2>\d+) | (?P<N>\d+) (?!\s*-))    """ class Graph:     def __init__(self, name, connections):        self.name = name        self.connections = connections        g = self.graph = defaultdict(list)  # maps vertex to direct connections         matches = re.finditer(connection_re, connections,                              re.MULTILINE | re.VERBOSE)        for match in matches:            n1, n2, n = match.groups()            if n:                g[n] += []            else:                g[n1].append(n2)    # Each the neighbour of the other                g[n2].append(n1)     def greedy_colour(self, order=None):        "Greedy colourisation algo."        if order is None:            order = self.graph      # Choose something        colour = self.colour = {}        neighbours = self.graph        for node in order:            used_neighbour_colours = (colour[nbr] for nbr in neighbours[node]                                      if nbr in colour)            colour[node] = first_avail_int(used_neighbour_colours)        self.pp_colours()        return colour     def pp_colours(self):        print(f"\n{self.name}")        c = self.colour        e = canonical_edges = set()        for n1, neighbours in sorted(self.graph.items()):            if neighbours:                for n2 in neighbours:                    edge = tuple(sorted([n1, n2]))                    if edge not in canonical_edges:                        print(f"       {n1}-{n2}: Colour: {c[n1]}, {c[n2]}")                        canonical_edges.add(edge)            else:                print(f"         {n1}: Colour: {c[n1]}")        lc = len(set(c.values()))        print(f"    #Nodes: {len(c)}\n    #Edges: {len(e)}\n  #Colours: {lc}")  def first_avail_int(data):    "return lowest int 0... not in data"    d = set(data)    for i in count():        if i not in d:            return i  if __name__ == '__main__':    for name, connections in [            ('Ex1', "0-1 1-2 2-0 3"),            ('Ex2', "1-6 1-7 1-8 2-5 2-7 2-8 3-5 3-6 3-8 4-5 4-6 4-7"),            ('Ex3', "1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6"),            ('Ex4', "1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7"),            ]:        g = Graph(name, connections)        g.greedy_colour()`
Output:
```Ex1
0-1: Colour: 0, 1
0-2: Colour: 0, 2
1-2: Colour: 1, 2
3: Colour: 0
#Nodes: 4
#Edges: 3
#Colours: 3

Ex2
1-6: Colour: 0, 1
1-7: Colour: 0, 1
1-8: Colour: 0, 1
2-5: Colour: 0, 1
2-7: Colour: 0, 1
2-8: Colour: 0, 1
3-5: Colour: 0, 1
3-6: Colour: 0, 1
3-8: Colour: 0, 1
4-5: Colour: 0, 1
4-6: Colour: 0, 1
4-7: Colour: 0, 1
#Nodes: 8
#Edges: 12
#Colours: 2

Ex3
1-4: Colour: 0, 1
1-6: Colour: 0, 1
1-8: Colour: 0, 1
2-3: Colour: 1, 0
2-5: Colour: 1, 0
2-7: Colour: 1, 0
3-6: Colour: 0, 1
3-8: Colour: 0, 1
4-5: Colour: 1, 0
4-7: Colour: 1, 0
5-8: Colour: 0, 1
6-7: Colour: 1, 0
#Nodes: 8
#Edges: 12
#Colours: 2

Ex4
1-6: Colour: 0, 1
1-7: Colour: 0, 1
1-8: Colour: 0, 1
2-5: Colour: 2, 0
2-7: Colour: 2, 1
2-8: Colour: 2, 1
3-5: Colour: 2, 0
3-6: Colour: 2, 1
3-8: Colour: 2, 1
4-5: Colour: 2, 0
4-6: Colour: 2, 1
4-7: Colour: 2, 1
#Nodes: 8
#Edges: 12
#Colours: 3```

Python dicts preserve insertion order and Ex2/Ex3 edges are traced in a similar way which could be the cause of exactly the same colours used for Ex2 and Ex3. The wp article must use an earlier version of Python/different ordering of edge definitions.

Ex4 changes the order of nodes enough to affect the number of colours used.

## Raku

(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```

## zkl

`fcn colorGraph(nodeStr){	// "0-1 1-2 2-0 3"   numEdges,graph := 0,Dictionary();  // ( 0:(1,2), 1:L(0,2), 2:(1,0), 3:() )   foreach n in (nodeStr.split(" ")){ // parse string to graph      n=n - " ";      if(n.holds("-")){	 a,b := n.split("-");	// keep as string	 graph.appendV(a,b); graph.appendV(b,a); 	 numEdges+=1;      }      else graph[n]=T;		// island   }   colors,colorPool := Dictionary(), ["A".."Z"].walk();   graph.pump(Void,'wrap([(node,nbrs)]){  // ( "1",(0,2), "3",() )      clrs:=colorPool.copy();	// all colors are available, then remove neighbours      foreach i in (nbrs){ clrs.remove(colors.find(i)) }  // if nbr has color, color not available      colors[node] = clrs[0];	// first available remaining color   });   return(graph,colors,numEdges)}`
`fcn printColoredGraph(graphStr){   graph,colors,numEdges := colorGraph(graphStr);   nodes:=graph.keys.sort();   println("Graph: ",graphStr);   println("Node/color: ",       nodes.pump(List,'wrap(v){ String(v,"/",colors[v]) }).concat(", "));   println("Node : neighbours --> colors:");   foreach node in (nodes){      ns:=graph[node];      println(node," : ",ns.concat(" "),"  -->  ",              colors[node]," : ",ns.apply(colors.get).concat(" "));   }   println("Number nodes:  ",nodes.len());   println("Number edges:  ",numEdges);   println("Number colors: ",       colors.values.pump(Dictionary().add.fp1(Void)).len());	// create set, count   println();}`
`graphs:=T(   "0-1 1-2 2-0 3",    "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");graphs.apply2(printColoredGraph);`
Output:
```Graph: 0-1 1-2 2-0 3
Node/color: 0/A, 1/B, 2/C, 3/A
Node : neighbours --> colors:
0 : 1 2  -->  A : B C
1 : 0 2  -->  B : A C
2 : 1 0  -->  C : B A
3 :   -->  A :
Number nodes:  4
Number edges:  3
Number colors: 3

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
Node/color: 1/A, 2/A, 3/A, 4/A, 5/B, 6/B, 7/B, 8/B
Node : neighbours --> colors:
1 : 6 7 8  -->  A : B B B
2 : 5 7 8  -->  A : B B B
3 : 5 6 8  -->  A : B B B
4 : 5 6 7  -->  A : B B B
5 : 2 3 4  -->  B : A A A
6 : 1 3 4  -->  B : A A A
7 : 1 2 4  -->  B : A A A
8 : 1 2 3  -->  B : A A A
Number nodes:  8
Number edges:  12
Number colors: 2

Graph: 1-4 1-6 1-8 3-2 3-6 3-8 5-2 5-4 5-8 7-2 7-4 7-6
Node/color: 1/A, 2/A, 3/B, 4/B, 5/C, 6/C, 7/D, 8/D
Node : neighbours --> colors:
1 : 4 6 8  -->  A : B C D
2 : 3 5 7  -->  A : B C D
3 : 2 6 8  -->  B : A C D
4 : 1 5 7  -->  B : A C D
5 : 2 4 8  -->  C : A B D
6 : 1 3 7  -->  C : A B D
7 : 2 4 6  -->  D : A B C
8 : 1 3 5  -->  D : A B C
Number nodes:  8
Number edges:  12
Number colors: 4

Graph: 1-6 7-1 8-1 5-2 2-7 2-8 3-5 6-3 3-8 4-5 4-6 4-7
Node/color: 1/A, 2/A, 3/A, 4/A, 5/B, 6/B, 7/B, 8/B
Node : neighbours --> colors:
1 : 6 7 8  -->  A : B B B
2 : 5 7 8  -->  A : B B B
3 : 5 6 8  -->  A : B B B
4 : 5 6 7  -->  A : B B B
5 : 2 3 4  -->  B : A A A
6 : 1 3 4  -->  B : A A A
7 : 1 2 4  -->  B : A A A
8 : 1 2 3  -->  B : A A A
Number nodes:  8
Number edges:  12
Number colors: 2
```