# Graph coloring¶

This module gathers all methods related to graph coloring. Here is what it can do :

Proper vertex coloring

 all_graph_colorings() Computes all $$n$$-colorings a graph first_coloring() Returns the first vertex coloring found number_of_n_colorings() Computes the number of $$n$$-colorings of a graph numbers_of_colorings() Computes the number of colorings of a graph chromatic_number() Returns the chromatic number of the graph vertex_coloring() Computes Vertex colorings and chromatic numbers

Other colorings

 grundy_coloring() Computes Grundy numbers and Grundy colorings b_coloring() Computes a b-chromatic numbers and b-colorings edge_coloring() Compute chromatic index and edge colorings round_robin() Computes a round-robin coloring of the complete graph on $$n$$ vertices linear_arboricity() Computes the linear arboricity of the given graph acyclic_edge_coloring() Computes an acyclic edge coloring of the current graph

AUTHORS:

• Tom Boothby (2008-02-21): Initial version
• Carlo Hamalainen (2009-03-28): minor change: switch to C++ DLX solver
• Nathann Cohen (2009-10-24): Coloring methods using linear programming

## Methods¶

class sage.graphs.graph_coloring.Test

This class performs randomized testing for all_graph_colorings. Since everything else in this file is derived from all_graph_colorings, this is a pretty good randomized tester for the entire file. Note that for a graph $$G$$, G.chromatic_polynomial() uses an entirely different algorithm, so we provide a good, independent test.

random(tests=1000)

Calls self.random_all_graph_colorings(). In the future, if other methods are added, it should call them, too.

TESTS:

sage: from sage.graphs.graph_coloring import Test
sage: Test().random(1)
random_all_graph_colorings(tests=1000)

Verifies the results of all_graph_colorings() in three ways:

1. all colorings are unique
2. number of m-colorings is $$P(m)$$ (where $$P$$ is the chromatic polynomial of the graph being tested)
3. colorings are valid – that is, that no two vertices of the same color share an edge.

TESTS:

sage: from sage.graphs.graph_coloring import Test
sage: Test().random_all_graph_colorings(1)
sage.graphs.graph_coloring.acyclic_edge_coloring(g, hex_colors=False, value_only=False, k=0, solver=None, verbose=0)

Computes an acyclic edge coloring of the current graph.

An edge coloring of a graph is a assignment of colors to the edges of a graph such that :

• the coloring is proper (no adjacent edges share a color)
• For any two colors $$i,j$$, the union of the edges colored with $$i$$ or $$j$$ is a forest.

The least number of colors such that such a coloring exists for a graph $$G$$ is written $$\chi'_a(G)$$, also called the acyclic chromatic index of $$G$$.

It is conjectured that this parameter can not be too different from the obvious lower bound $$\Delta(G)\leq \chi'_a(G)$$, $$\Delta(G)$$ being the maximum degree of $$G$$, which is given by the first of the two constraints. Indeed, it is conjectured that $$\Delta(G)\leq \chi'_a(G) \leq \Delta(G) + 2$$.

INPUT:

• hex_colors (boolean)

• If hex_colors = True, the function returns a dictionary associating to each color a list of edges (meant as an argument to the edge_colors keyword of the plot method).
• If hex_colors = False (default value), returns a list of graphs corresponding to each color class.
• value_only (boolean)

• If value_only = True, only returns the acyclic chromatic index as an integer value
• If value_only = False, returns the color classes according to the value of hex_colors
• k (integer) – the number of colors to use.

• If k>0, computes an acyclic edge coloring using $$k$$ colors.
• If k=0 (default), computes a coloring of $$G$$ into $$\Delta(G) + 2$$ colors, which is the conjectured general bound.
• If k=None, computes a decomposition using the least possible number of colors.
• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.

• verbose – integer (default: 0). Sets the level of

verbosity. Set to 0 by default, which means quiet.

ALGORITHM:

Linear Programming

EXAMPLE:

The complete graph on 8 vertices can not be acyclically edge-colored with less $$\Delta+1$$ colors, but it can be colored with $$\Delta+2=9$$:

sage: from sage.graphs.graph_coloring import acyclic_edge_coloring
sage: g = graphs.CompleteGraph(8)
sage: colors = acyclic_edge_coloring(g)

Each color class is of course a matching

sage: all([max(gg.degree())<=1 for gg in colors])
True

These matchings being a partition of the edge set:

sage: all([ any([gg.has_edge(e) for gg in colors]) for e in g.edges(labels = False)])
True

Besides, the union of any two of them is a forest

sage: all([g1.union(g2).is_forest() for g1 in colors for g2 in colors])
True

If one wants to acyclically color a cycle on $$4$$ vertices, at least 3 colors will be necessary. The function raises an exception when asked to color it with only 2:

sage: g = graphs.CycleGraph(4)
sage: acyclic_edge_coloring(g, k=2)
Traceback (most recent call last):
...
ValueError: This graph can not be colored with the given number of colors.

The optimal coloring give us $$3$$ classes:

sage: colors = acyclic_edge_coloring(g, k=None)
sage: len(colors)
3
sage.graphs.graph_coloring.all_graph_colorings(G, n, count_only=False, hex_colors=False, vertex_color_dict=False)

Computes all $$n$$-colorings of the graph $$G$$ by casting the graph coloring problem into an exact cover problem, and passing this into an implementation of the Dancing Links algorithm described by Knuth (who attributes the idea to Hitotumatu and Noshita).

INPUT:

• G - a graph

• n - a positive integer the number of colors

• $$count_only$$ – (default: False) when set to True, it returns 1

for each coloring

• $$hex_colors$$ – (default: False) when set to False, it labels the colors [0,1,..,n-1], otherwise it uses the RGB Hex labeling

• $$vertex_color_dict$$ – (default: False) when set to True, it returns a dictionary {vertex:color}, otherwise it returns a dictionary {color:[list of vertices]}

The construction works as follows. Columns:

• The first $$|V|$$ columns correspond to a vertex – a $$1$$ in this column indicates that that vertex has a color.
• After those $$|V|$$ columns, we add $$n*|E|$$ columns – a $$1$$ in these columns indicate that a particular edge is incident to a vertex with a certain color.

Rows:

• For each vertex, add $$n$$ rows; one for each color $$c$$. Place a $$1$$ in the column corresponding to the vertex, and a $$1$$ in the appropriate column for each edge incident to the vertex, indicating that that edge is incident to the color $$c$$.
• If $$n > 2$$, the above construction cannot be exactly covered since each edge will be incident to only two vertices (and hence two colors) - so we add $$n*|E|$$ rows, each one containing a $$1$$ for each of the $$n*|E|$$ columns. These get added to the cover solutions “for free” during the backtracking.

Note that this construction results in $$n*|V| + 2*n*|E| + n*|E|$$ entries in the matrix. The Dancing Links algorithm uses a sparse representation, so if the graph is simple, $$|E| \leq |V|^2$$ and $$n <= |V|$$, this construction runs in $$O(|V|^3)$$ time. Back-conversion to a coloring solution is a simple scan of the solutions, which will contain $$|V| + (n-2)*|E|$$ entries, so runs in $$O(|V|^3)$$ time also. For most graphs, the conversion will be much faster – for example, a planar graph will be transformed for $$4$$-coloring in linear time since $$|E| = O(|V|)$$.

REFERENCES:

http://www-cs-staff.stanford.edu/~uno/papers/dancing-color.ps.gz

EXAMPLES:

sage: from sage.graphs.graph_coloring import all_graph_colorings
sage: G = Graph({0:[1,2,3],1:[2]})
sage: n = 0
sage: for C in all_graph_colorings(G,3,hex_colors=True):
...       parts = [C[k] for k in C]
...       for P in parts:
...           l = len(P)
...           for i in range(l):
...               for j in range(i+1,l):
...                   if G.has_edge(P[i],P[j]):
...                       raise RuntimeError, "Coloring Failed."
...       n+=1
sage: print "G has %s 3-colorings."%n
G has 12 3-colorings.

TESTS:

sage: G = Graph({0:[1,2,3],1:[2]})
sage: for C in all_graph_colorings(G,0): print C
sage: for C in all_graph_colorings(G,-1): print C
Traceback (most recent call last):
...
ValueError: n must be non-negative.
sage: G = Graph({0:[1],1:[2]})
sage: for c in all_graph_colorings(G,2, vertex_color_dict = True): print c
{0: 0, 1: 1, 2: 0}
{0: 1, 1: 0, 2: 1}
sage: for c in all_graph_colorings(G,2,hex_colors = True): print c
{'#00ffff': [1], '#ff0000': [0, 2]}
{'#ff0000': [1], '#00ffff': [0, 2]}
sage: for c in all_graph_colorings(G,2,hex_colors=True,vertex_color_dict = True): print c
{0: '#ff0000', 1: '#00ffff', 2: '#ff0000'}
{0: '#00ffff', 1: '#ff0000', 2: '#00ffff'}
sage: for c in all_graph_colorings(G, 2, vertex_color_dict = True): print c
{0: 0, 1: 1, 2: 0}
{0: 1, 1: 0, 2: 1}
sage: for c in all_graph_colorings(G, 2, count_only=True, vertex_color_dict = True): print c
1
1
sage.graphs.graph_coloring.b_coloring(g, k, value_only=True, solver=None, verbose=0)

Computes a b-coloring with at most k colors that maximizes the number of colors, if such a coloring exists

Definition :

Given a proper coloring of a graph $$G$$ and a color class $$C$$ such that none of its vertices have neighbors in all the other color classes, one can eliminate color class $$C$$ assigning to each of its elements a missing color in its neighborhood.

Let a b-vertex be a vertex with neighbors in all other colorings. Then, one can repeat the above procedure until a coloring is obtained where every color class contains a b-vertex, in which case none of the color classes can be eliminated with the same ideia. So, one can define a b-coloring as a proper coloring where each color class has a b-vertex.

In the worst case, after successive applications of the above procedure, one get a proper coloring that uses a number of colors equal to the the b-chromatic number of $$G$$ (denoted $$\chi_b(G)$$): the maximum $$k$$ such that $$G$$ admits a b-coloring with $$k$$ colors.

An useful upper bound for calculating the b-chromatic number is the following. If G admits a b-coloring with k colors, then there are $$k$$ vertices of degree at least $$k - 1$$ (the b-vertices of each color class). So, if we set $$m(G) = max$$ {$$k | there are k vertices of degree at least k - 1$$}, we have that $$\chi_b(G) \leq m(G)$$.

Note

This method computes a b-coloring that uses at MOST $$k$$ colors. If this method returns a value equal to $$k$$, it can not be assumed that $$k$$ is equal to $$\chi_b(G)$$. Meanwhile, if it returns any value $$k' < k$$, this is a certificate that the Grundy number of the given graph is $$k'$$.

As $$\chi_b(G)\leq m(G)$$, it can be assumed that $$\chi_b(G) = k$$ if b_coloring(g, k) returns $$k$$ when $$k = m(G)$$.

INPUT:

• k (integer) – Maximum number of colors
• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
• value_only – boolean (default: True). When set to True, only the number of colors is returned. Otherwise, the pair (nb_colors, coloring) is returned, where coloring is a dictionary associating its color (integer) to each vertex of the graph.
• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

ALGORITHM:

Integer Linear Program.

EXAMPLES:

The b-chromatic number of a $$P_5$$ is equal to 3:

sage: from sage.graphs.graph_coloring import b_coloring
sage: g = graphs.PathGraph(5)
sage: b_coloring(g, 5)
3

The b-chromatic number of the Petersen Graph is equal to 3:

sage: g = graphs.PetersenGraph()
sage: b_coloring(g, 5)
3

It would have been sufficient to set the value of k to 4 in this case, as $$4 = m(G)$$.

sage.graphs.graph_coloring.chromatic_number(G)

Returns the minimal number of colors needed to color the vertices of the graph $$G$$.

EXAMPLES:

sage: from sage.graphs.graph_coloring import chromatic_number
sage: G = Graph({0:[1,2,3],1:[2]})
sage: chromatic_number(G)
3

sage: G = graphs.PetersenGraph()
sage: G.chromatic_number()
3
sage.graphs.graph_coloring.edge_coloring(g, value_only=False, vizing=False, hex_colors=False, solver=None, verbose=0)

Properly colors the edges of a graph. See the URL http://en.wikipedia.org/wiki/Edge_coloring for further details on edge coloring.

INPUT:

• g – a graph.

• value_only – (default: False):

• When set to True, only the chromatic index is returned.
• When set to False, a partition of the edge set into matchings is returned if possible.
• vizing – (default: False):

• When set to True, tries to find a $$\Delta + 1$$-edge-coloring, where $$\Delta$$ is equal to the maximum degree in the graph.
• When set to False, tries to find a $$\Delta$$-edge-coloring, where $$\Delta$$ is equal to the maximum degree in the graph. If impossible, tries to find and returns a $$\Delta + 1$$-edge-coloring. This implies that value_only=False.
• hex_colors – (default: False) when set to True, the partition returned is a dictionary whose keys are colors and whose values are the color classes (ideal for plotting).

• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.

• verbose – integer (default: 0). Sets the level of

verbosity. Set to 0 by default, which means quiet.

OUTPUT:

In the following, $$\Delta$$ is equal to the maximum degree in the graph g.

• If vizing=True and value_only=False, return a partition of the edge set into $$\Delta + 1$$ matchings.
• If vizing=False and value_only=True, return the chromatic index.
• If vizing=False and value_only=False, return a partition of the edge set into the minimum number of matchings.
• If vizing=True and value_only=True, should return something, but mainly you are just trying to compute the maximum degree of the graph, and this is not the easiest way. By Vizing’s theorem, a graph has a chromatic index equal to $$\Delta$$ or to $$\Delta + 1$$.

Note

In a few cases, it is possible to find very quickly the chromatic index of a graph, while it remains a tedious job to compute a corresponding coloring. For this reason, value_only = True can sometimes be much faster, and it is a bad idea to compute the whole coloring if you do not need it !

EXAMPLE:

sage: from sage.graphs.graph_coloring import edge_coloring
sage: g = graphs.PetersenGraph()
sage: edge_coloring(g, value_only=True)
4

Complete graphs are colored using the linear-time round-robin coloring:

sage: from sage.graphs.graph_coloring import edge_coloring
sage: len(edge_coloring(graphs.CompleteGraph(20)))
19
sage.graphs.graph_coloring.first_coloring(G, n=0, hex_colors=False)

Given a graph, and optionally a natural number $$n$$, returns the first coloring we find with at least $$n$$ colors.

INPUT:

• hex_colors – (default: False) when set to True, the partition returned is a dictionary whose keys are colors and whose values are the color classes (ideal for plotting).
• n – The minimal number of colors to try.

EXAMPLES:

sage: from sage.graphs.graph_coloring import first_coloring
sage: G = Graph({0: [1, 2, 3], 1: [2]})
sage: first_coloring(G, 3)
[[1, 3], [0], [2]]
sage.graphs.graph_coloring.grundy_coloring(g, k, value_only=True, solver=None, verbose=0)

Computes the worst-case of a first-fit coloring with less than $$k$$ colors.

Definition :

A first-fit coloring is obtained by sequentially coloring the vertices of a graph, assigning them the smallest color not already assigned to one of its neighbors. The result is clearly a proper coloring, which usually requires much more colors than an optimal vertex coloring of the graph, and heavily depends on the ordering of the vertices.

The number of colors required by the worst-case application of this algorithm on a graph $$G$$ is called the Grundy number, written $$\Gamma (G)$$.

Equivalent formulation :

Equivalently, a Grundy coloring is a proper vertex coloring such that any vertex colored with $$i$$ has, for every $$j<i$$, a neighbor colored with $$j$$. This can define a Linear Program, which is used here to compute the Grundy number of a graph.

INPUT:

• k (integer) – Maximum number of colors
• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
• value_only – boolean (default: True). When set to True, only the number of colors is returned. Otherwise, the pair (nb_colors, coloring) is returned, where coloring is a dictionary associating its color (integer) to each vertex of the graph.
• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

ALGORITHM:

Integer Linear Program.

EXAMPLES:

The Grundy number of a $$P_4$$ is equal to 3:

sage: from sage.graphs.graph_coloring import grundy_coloring
sage: g = graphs.PathGraph(4)
sage: grundy_coloring(g, 4)
3

The Grundy number of the PetersenGraph is equal to 4:

sage: g = graphs.PetersenGraph()
sage: grundy_coloring(g, 5)
4

It would have been sufficient to set the value of k to 4 in this case, as $$4 = \Delta(G)+1$$.

sage.graphs.graph_coloring.linear_arboricity(g, plus_one=None, hex_colors=False, value_only=False, solver=None, verbose=0)

Computes the linear arboricity of the given graph.

The linear arboricity of a graph $$G$$ is the least number $$la(G)$$ such that the edges of $$G$$ can be partitioned into linear forests (i.e. into forests of paths).

Obviously, $$la(G)\geq \lceil \frac {\Delta(G)} 2 \rceil$$.

It is conjectured in [Aki80] that $$la(G)\leq \lceil \frac {\Delta(G)+1} 2 \rceil$$.

INPUT:

• hex_colors (boolean)

• If hex_colors = True, the function returns a dictionary associating to each color a list of edges (meant as an argument to the edge_colors keyword of the plot method).
• If hex_colors = False (default value), returns a list of graphs corresponding to each color class.
• value_only (boolean)

• If value_only = True, only returns the linear arboricity as an integer value.
• If value_only = False, returns the color classes according to the value of hex_colors
• plus_one (integer) – whether to use $$\lceil \frac {\Delta(G)} 2 \rceil$$ or $$\lceil \frac {\Delta(G)+1} 2 \rceil$$ colors.

• If 0, computes a decomposition of $$G$$ into $$\lceil \frac {\Delta(G)} 2 \rceil$$ forests of paths
• If 1, computes a decomposition of $$G$$ into $$\lceil \frac {\Delta(G)+1} 2 \rceil$$ colors, which is the conjectured general bound.
• If plus_one = None (default), computes a decomposition using the least possible number of colors.
• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.

• verbose – integer (default: 0). Sets the level of verbosity. Set

to 0 by default, which means quiet.

ALGORITHM:

Linear Programming

COMPLEXITY:

NP-Hard

EXAMPLE:

Obviously, a square grid has a linear arboricity of 2, as the set of horizontal lines and the set of vertical lines are an admissible partition:

sage: from sage.graphs.graph_coloring import linear_arboricity
sage: g = graphs.GridGraph([4,4])
sage: g1,g2 = linear_arboricity(g)

Each graph is of course a forest:

sage: g1.is_forest() and g2.is_forest()
True

Of maximum degree 2:

sage: max(g1.degree()) <= 2 and max(g2.degree()) <= 2
True

Which constitutes a partition of the whole edge set:

sage: all([g1.has_edge(e) or g2.has_edge(e) for e in g.edges(labels = None)])
True

REFERENCES:

 [Aki80] Akiyama, J. and Exoo, G. and Harary, F. Covering and packing in graphs. III: Cyclic and acyclic invariants Mathematical Institute of the Slovak Academy of Sciences Mathematica Slovaca vol30, n4, pages 405–417, 1980
sage.graphs.graph_coloring.number_of_n_colorings(G, n)

Computes the number of $$n$$-colorings of a graph

EXAMPLES:

sage: from sage.graphs.graph_coloring import number_of_n_colorings
sage: G = Graph({0:[1,2,3],1:[2]})
sage: number_of_n_colorings(G,3)
12
sage.graphs.graph_coloring.numbers_of_colorings(G)

Returns the number of $$n$$-colorings of the graph $$G$$ for $$n$$ from $$0$$ to $$|V|$$.

EXAMPLES:

sage: from sage.graphs.graph_coloring import numbers_of_colorings
sage: G = Graph({0:[1,2,3],1:[2]})
sage: numbers_of_colorings(G)
[0, 0, 0, 12, 72]
sage.graphs.graph_coloring.round_robin(n)

Computes a round-robin coloring of the complete graph on $$n$$ vertices.

A round-robin coloring of the complete graph $$G$$ on $$2n$$ vertices ($$V = [0, \dots, 2n - 1]$$) is a proper coloring of its edges such that the edges with color $$i$$ are all the $$(i + j, i - j)$$ plus the edge $$(2n - 1, i)$$.

If $$n$$ is odd, one obtain a round-robin coloring of the complete graph through the round-robin coloring of the graph with $$n + 1$$ vertices.

INPUT:

• n – the number of vertices in the complete graph.

OUTPUT:

• A CompleteGraph with labelled edges such that the label of each edge is its color.

EXAMPLES:

sage: from sage.graphs.graph_coloring import round_robin
sage: round_robin(3).edges()
[(0, 1, 2), (0, 2, 1), (1, 2, 0)]
sage: round_robin(4).edges()
[(0, 1, 2), (0, 2, 1), (0, 3, 0), (1, 2, 0), (1, 3, 1), (2, 3, 2)]

For higher orders, the coloring is still proper and uses the expected number of colors.

sage: g = round_robin(9)
sage: sum([Set([e[2] for e in g.edges_incident(v)]).cardinality() for v in g]) == 2*g.size()
True
sage: Set([e[2] for e in g.edge_iterator()]).cardinality()
9
sage: g = round_robin(10)
sage: sum([Set([e[2] for e in g.edges_incident(v)]).cardinality() for v in g]) == 2*g.size()
True
sage: Set([e[2] for e in g.edge_iterator()]).cardinality()
9
sage.graphs.graph_coloring.vertex_coloring(g, k=None, value_only=False, hex_colors=False, solver=None, verbose=0)

Computes the chromatic number of the given graph or tests its $$k$$-colorability. See http://en.wikipedia.org/wiki/Graph_coloring for further details on graph coloring.

INPUT:

• g – a graph.

• k – (default: None) tests whether the graph is $$k$$-colorable. The function returns a partition of the vertex set in $$k$$ independent sets if possible and False otherwise.

• value_only – (default: False):

• When set to True, only the chromatic number is returned.
• When set to False (default), a partition of the vertex set into independent sets is returned if possible.
• hex_colors – (default: False) when set to True, the partition returned is a dictionary whose keys are colors and whose values are the color classes (ideal for plotting).

• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.

• verbose – integer (default: 0). Sets the level of

verbosity. Set to 0 by default, which means quiet.

OUTPUT:

• If k=None and value_only=False, then return a partition of the vertex set into the minimum possible of independent sets.
• If k=None and value_only=True, return the chromatic number.
• If k is set and value_only=None, return False if the graph is not $$k$$-colorable, and a partition of the vertex set into $$k$$ independent sets otherwise.
• If k is set and value_only=True, test whether the graph is $$k$$-colorable, and return True or False accordingly.

EXAMPLE:

sage: from sage.graphs.graph_coloring import vertex_coloring
sage: g = graphs.PetersenGraph()
sage: vertex_coloring(g, value_only=True)
3