DiscreteMath`Combinatorica` extends Mathematica by over 450 functions in combinatorics and graph theory. It includes functions for constructing graphs and other combinatorial objects, computing invariants of these objects, and finally displaying them. This documentation covers only a subset of these functions. The best guide to this package is the book Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica, by Steven Skiena and Sriram Pemmaraju, published by Cambridge University Press, 2003. The new Combinatorica is a substantial rewrite of the original 1990 version. It is now much faster than before, and provides improved graphics and significant additional functionality.

We encourage you to visit our website, www.combinatorica.com, where you will find the latest release of the package, an editor for Combinatorica graphs, and additional files of interest.

This loads the package.



Permutations and Combinations

Permutations and subsets are the most basic combinatorial objects. DiscreteMath`Combinatorica` provides functions for constructing objects both randomly and deterministically, to rank and unrank them, and to compute invariants on them. Here we provide examples of some of these functions in action.

These permutations are generated in minimum change order, where successive permutations differ by exactly one transposition. The built-in generator Permutations constructs permutations in lexicographic order.




{{a, b, c}, {b, a, c}, {c, a, b}, {a, c, b}, {b, c, a}, {c, b, a}}

The ranking function illustrates that the built-in function Permutations uses lexicographic sequencing.


Map[RankPermutation, Permutations[{1,2,3,4}]]


{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}

With (3 !) = 6 distinct permutations of three elements, within 20 random permutations we are likely to see all of them. Observe that it is unlikely for the first six permutations to all be distinct.


Table[RandomPermutation[3], {20}]


{{3, 1, 2}, {1, 2, 3}, {1, 3, 2}, {1, 2, 3}, {3, 2, 1}, {2, 3, 1}, {2, 1, 3}, {3, 2, 1}, {1, 3 ... {3, 1, 2}, {2, 3, 1}, {3, 2, 1}, {2, 1, 3}, {2, 3, 1}, {1, 2, 3}, {1, 3, 2}, {3, 2, 1}, {3, 2, 1}}

A fixed point of a permutation p is an element in the same position in p as in the inverse of p. Thus, the only fixed point in this permutation is 7.




{5, 4, 6, 1, 3, 8, 7, 2}

The identity permutation consists of n singleton cycles or fixed points.




{{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {10}}

The classic problem in Polya theory is counting how many different ways necklaces can be made out of k beads, when there are m different types or colors of beads to choose from. When two necklaces are considered the same if they can be obtained only by shifting the beads (as opposed to turning the necklace over), the symmetries are defined by k permutations, each of which is a cyclic shift of the identity permutation. When a variable is specified for the number of colors, a polynomial results.




m/2 + m^2/4 + m^4/8 + m^8/8

The number of inversions in a permutation is equal to that of its inverse.


(p=RandomPermutation[50]; {Inversions[p], Inversions[InversePermutation[p]]})


{642, 642}

Generating subsets incrementally is efficient when the goal is to find the first subset with a given property, since every subset need not be constructed.


Table[UnrankSubset[n,{a,b,c,d}], {n,0,15}]


{{}, {d}, {c, d}, {c}, {b, c}, {b, c, d}, {b, d}, {b}, {a, b}, {a, b, d}, {a, b, c, d}, {a, b, c}, {a, c}, {a, c, d}, {a, d}, {a}}

In a Gray code, each subset differs in exactly one element from its neighbors. Observe that the last eight subsets all contain 1, while none of the first eight do.




{{}, {4}, {3, 4}, {3}, {2, 3}, {2, 3, 4}, {2, 4}, {2}, {1, 2}, {1, 2, 4}, {1, 2, 3, 4}, {1, 2, 3}, {1, 3}, {1, 3, 4}, {1, 4}, {1}}

A k-subset is a subset with exactly k elements in it. Since the lead element is placed in first, the k-subsets are given in lexicographic order.




{{1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5}, {1, 4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5}, {3, 4, 5}}

BinarySearch DerangementQ
Derangements DistinctPermutations
EncroachingListSet FromCycles
FromInversionVector HeapSort
Heapify HideCycles
IdentityPermutation Index
InversePermutation InversionPoset
Inversions InvolutionQ
Involutions Josephus
LexicographicPermutations LongestIncreasingSubsequence
MinimumChangePermutations NextPermutation
PermutationQ PermutationType
PermutationWithCycle Permute
RandomHeap RandomPermutation
RankPermutation RevealCycles
Runs SelectionSort
SignaturePermutation ToCycles
ToInversionVector UnrankPermutation

Combinatorica functions for permutations.

BinarySubsets DeBruijnSequence
GrayCodeKSubsets GrayCodeSubsets
GrayGraph KSubsets
LexicographicSubsets NextBinarySubset
NextGrayCodeSubset NextKSubset
NextLexicographicSubset NextSubset
NthSubset RandomKSubset
RandomSubset RankBinarySubset
RankGrayCodeSubset RankKSubset
RankSubset Strings
Subsets UnrankBinarySubset
UnrankGrayCodeSubset UnrankKSubset

Combinatorica functions for subsets.

AlternatingGroup AlternatingGroupIndex
CycleIndex CycleStructure
Cycles Cyclic
CyclicGroup CyclicGroupIndex
Dihedral DihedralGroup
DihedralGroupIndex EquivalenceClasses
KSubsetGroup KSubsetGroupIndex
ListNecklaces MultiplicationTable
NecklacePolynomial OrbitInventory
OrbitRepresentatives Orbits
Ordered PairGroup
PairGroupIndex PermutationGroupQ
SamenessRelation SymmetricGroup

Combinatorica functions for group theory.

Partitions, Compositions, and Young Tableaux

A partition of a positive integer n is a set of k strictly positive integers whose sum is n. A composition of n is a particular arrangement of nonnegative integers whose sum is n. A set partition of n elements is a grouping of all the elements into nonempty, nonintersecting subsets. A Young tableau is a structure of integers 1, …, n where the number of elements in each row is defined by an integer partition of n. Further, the elements of each row and column are in increasing order, and the rows are left justified. These four related combinatorial objects have a host of interesting applications and properties.

Here are the eleven partitions of 6. Observe that they are given in reverse lexicographic order.




{{6}, {5, 1}, {4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1, 1}, {2, 2, 2}, {2, 2, 1, 1}, {2, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}}

Although the number of partitions grows exponentially, it does so more slowly than permutations or subsets, so complete tables can be generated for larger values of n.





Ferrers diagrams represent partitions as patterns of dots. They provide a useful tool for visualizing partitions, because moving the dots around provides a mechanism for proving bijections between classes of partitions. Here we construct a random partition of 100.






Here every composition of 5 into 3 parts is generated exactly once.




{{0, 0, 5}, {0, 1, 4}, {0, 2, 3}, {0, 3, 2}, {0, 4, 1}, {0, 5, 0}, {1, 0, 4}, {1, 1, 3}, {1, 2 ... {2, 1, 2}, {2, 2, 1}, {2, 3, 0}, {3, 0, 2}, {3, 1, 1}, {3, 2, 0}, {4, 0, 1}, {4, 1, 0}, {5, 0, 0}}

Set partitions are different than integer partitions, representing the ways we can partition distinct elements into subsets. They are useful for representing colorings and clusterings.




{{{1, 2, 3}}, {{1}, {2, 3}}, {{1, 2}, {3}}, {{1, 3}, {2}}, {{1}, {2}, {3}}}

The list of tableaux of shape {2, 2, 1} illustrates the amount of freedom available to tableaux structures. The smallest element is always in the upper left-hand corner, but the largest element is free to be the rightmost position of the last row defined by the distinct parts of the partition.




{{{1, 4}, {2, 5}, {3}}, {{1, 3}, {2, 5}, {4}}, {{1, 2}, {3, 5}, {4}}, {{1, 3}, {2, 4}, {5}}, {{1, 2}, {3, 4}, {5}}}

By iterating through the different integer partitions as shapes, all tableaux of a particular size can be constructed.




{{{1, 2, 3}}, {{1, 3}, {2}}, {{1, 2}, {3}}, {{1}, {2}, {3}}}

The hook length formula can be used to count the number of tableaux for any shape. Using the hook length formula over all partitions of n computes the number of tableaux on n elements.





Each of the 117,123,756,750 tableaux of this shape will be selected with equal likelihood.


TableForm[ RandomTableau[{6,5,5,4,3,2}] ]


1 2 3 4 16 17
5 6 12 19 21
7 8 13 20 24
9 10 18 22
11 14 23
15 25

A pigeonhole result states that any sequence of n^2 + 1 distinct integers must contain either an increasing or a decreasing scattered subsequence of length n + 1.


        RandomPermutation[50] ]


{1, 6, 9, 11, 17, 19, 21, 25, 27, 33, 34, 35}

Compositions DominatingIntegerPartitionQ
DominationLattice DurfeeSquare
FerrersDiagram NextComposition
NextPartition PartitionQ
Partitions RandomComposition
RandomPartition TransposePartition

Combinatorica functions for integer partitions.

CoarserSetPartitionQ FindSet
InitializeUnionFind KSetPartitions
PartitionLattice RGFQ
RGFToSetPartition RGFs
RandomKSetPartition RandomRGF
RandomSetPartition RankKSetPartition
RankRGF RankSetPartition
SetPartitionListViaRGF SetPartitionQ
SetPartitionToRGF SetPartitions
ToCanonicalSetPartition UnionSet
UnrankKSetPartition UnrankRGF

Combinatorica functions for set partitions.

ConstructTableau DeleteFromTableau
FirstLexicographicTableau InsertIntoTableau
LastLexicographicTableau NextTableau
PermutationToTableaux RandomTableau
TableauClasses TableauQ
Tableaux TableauxToPermutation

Combinatorica functions for Young tableaux.

Backtrack BellB
Cofactor Distribution
Element Eulerian
NumberOf2Paths NumberOfCompositions
NumberOfDerangements NumberOfDirectedGraphs
NumberOfGraphs NumberOfInvolutions
NumberOfKPaths NumberOfNecklaces
NumberOfPartitions NumberOfPermutationsByCycles
NumberOfPermutationsByInversions NumberOfPermutationsByType
NumberOfSpanningTrees NumberOfTableaux
StirlingFirst StirlingSecond

Combinatorica functions for counting.

Representing Graphs

We define a graph to be a set of vertices with a set of edges, where an edge is defined as a pair of vertices. The representation of graphs takes on different requirements depending upon whether the intended consumer is a person or a machine. Computers digest graphs best as data structures such as adjacency matrices or lists. People prefer a visualization of the structure as a collection of points connected by lines, which implies adding geometric information to the graph.

In the complete graph on five vertices, denoted K_5, each vertex is adjacent to all other vertices. CompleteGraph[n] constructs the complete graph on n vertices.


ShowGraph[ CompleteGraph[5] ];


The internals of the graph representation are not shown to the user—only a notation with the number of edges and vertices, followed by whether the graph is directed or undirected.




⁃Graph:<10, 5, Undirected>⁃

The adjacency matrix of K_5 shows that each vertex is adjacent to all other vertices. The main diagonal consists of zeros, since there are no self-loops in the complete graph, meaning edges from a vertex to itself.




0 1 1 1 1
1 0 1 1 1
1 1 0 1 1
1 1 1 0 1
1 1 1 1 0

The standard embedding of K_5 consists of five vertices equally spaced on a circle.


Vertices[ CompleteGraph[5] ]


RowBox[{{, RowBox[{RowBox[{{, RowBox[{0.309017, ,, 0.951057}], }}], ,, RowBox[{{, RowBox[{RowB ... {, RowBox[{0.309017, ,, RowBox[{-, 0.951057}]}], }}], ,, RowBox[{{, RowBox[{1., ,, 0}], }}]}], }}]

The number of vertices in a graph is termed the order of the graph.


V[ CompleteGraph[5] ]



M returns the number of edges in a graph.


M[ CompleteGraph[5] ]



Edge/vertex colors/styles can be globally modified, giving complete flexibility to change the appearance of a graph.


g = SetGraphOptions[CompleteGraph[4], VertexColor -> Red,
     EdgeColor -> Blue]


⁃Graph:<6, 4, Undirected>⁃

The colors, styles, labels, and weights of individual vertices and edges can also be changed individually, perhaps to highlight interesting features of the graph.


ShowGraph[ SetGraphOptions[ CompleteGraph[4],
       {{1, 2, VertexColor -> Green, VertexStyle -> Disk[Large]},
        {3, 4, VertexColor -> Blue}},
       EdgeColor -> Red
] ];


A star is a tree with one vertex of degree n - 1. Adding any new edge to a star produces a cycle of length 3.


ShowGraph[ AddEdge[Star[10], {1,2}] ];


Graphs with multi-edges and self-loops are supported. Here there are two copies of each edge of a star.


ShowGraph[ GraphSum[Star[10], Star[10]] ];


The adjacency list representation of a graph consists of n lists, one list for each vertex v_i, 1≤i≤n, which records the vertices to which v_i is adjacent. Each vertex in the complete graph is adjacent to all other vertices.


TableForm[ ToAdjacencyLists[CompleteGraph[5]] ]


2 3 4 5
1 3 4 5
1 2 4 5
1 2 3 5
1 2 3 4

There are n (n - 1) ordered pairs of edges defined by a complete graph of order n.


ToOrderedPairs[ CompleteGraph[5] ]


{{2, 1}, {3, 1}, {4, 1}, {5, 1}, {3, 2}, {4, 2}, {5, 2}, {4, 3}, {5, 3}, {5, 4}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}}

An induced subgraph of a graph G is a subset of the vertices of G together with any edges whose endpoints are both in this subset. An induced subgraph that is complete is called a clique. Any subset of the vertices in a complete graph defines a clique.


ShowGraph[ InduceSubgraph[CompleteGraph[20],
        RandomSubset[Range[20]]] ];


The vertices of a bipartite graph have the property that they can be partitioned into two sets such that no edge connects two vertices of the same set. Contracting an edge in a bipartite graph can ruin its bipartiteness. Note the self-loop created by the contraction.


ShowGraph[ Contract[ CompleteGraph[6,6],{1,7} ] ];


A breadth-first search of a graph explores all the vertices adjacent to the current vertex before moving on. A breadth-first traversal of a simple cycle alternates sides as it wraps around the cycle.




{1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 19, 17, 15, 13, 11, 9, 7, 5, 3}

In a depth-first search, the children of the first son of a vertex are explored before visiting his brothers. The depth-first traversal differs from the breadth-first traversal above in that it proceeds directly around the cycle.


DepthFirstTraversal[Cycle[20], 1]


{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

Different drawings or embeddings of a graph can reveal different aspects of its structure. The default embedding for a grid graph is a ranked embedding from all the vertices on one side. Ranking from the center vertex yields a different but interesting drawing.




The radial embedding of a tree is guaranteed to be planar, but radial embeddings can be used with any graph. Here we see a radial embedding of a random labeled tree.


ShowGraph[ RandomTree[10] ];


By arbitrarily selecting a root, any tree can be represented as a rooted tree.


ShowGraph[ RootedEmbedding[RandomTree[10],1] ];


An interesting general heuristic for drawing graphs models the graph as a system of springs and lets Hooke's law space the vertices. Here it does a good job illustrating the Join operation, where each vertex of K_7 is connected to each of two disconnected vertices. In achieving the minimum energy configuration, these two vertices end up on different sides of K_7.


        GraphJoin[EmptyGraph[2], CompleteGraph[7]]]];


AddEdge AddEdges
AddVertex AddVertices
ChangeEdges ChangeVertices
Contract DeleteCycle
DeleteEdge DeleteEdges
DeleteVertex DeleteVertices
InduceSubgraph MakeDirected
MakeSimple MakeUndirected
PermuteSubgraph RemoveMultipleEdges
RemoveSelfLoops ReverseEdges

Combinatorica functions for modifying graphs.

Edges FromAdjacencyLists
FromAdjacencyMatrix FromOrderedPairs
FromUnorderedPairs IncidenceMatrix
ToAdjacencyLists ToAdjacencyMatrix
ToOrderedPairs ToUnorderedPairs

Combinatorica functions for graph format translation.

Algorithm Box
Brelaz Center
Circle Directed
Disk EdgeColor
EdgeDirection EdgeLabel
EdgeLabelColor EdgeLabelPosition
EdgeStyle EdgeWeight
Euclidean HighlightedEdgeColors
HighlightedEdgeStyle HighlightedVertexColors
HighlightedVertexStyle Invariants
LNorm Large
LoopPosition LowerLeft
LowerRight NoMultipleEdges
NoSelfLoops Normal
NormalDashed NormalizeVertices
One Optimum
Parent PlotRange
RandomInteger Simple
Small Strong
Thick ThickDashed
Thin ThinDashed
Type Undirected
UpperLeft UpperRight
VertexColor VertexLabel
VertexLabelColor VertexLabelPosition
VertexNumber VertexNumberColor
VertexNumberPosition VertexStyle
VertexWeight Weak
WeightRange WeightingFunction

Combinatorica options for graph functions.

GetEdgeLabels GetEdgeWeights
GetVertexLabels GetVertexWeights
SetEdgeLabels SetEdgeWeights
SetGraphOptions SetVertexLabels

Combinatorica functions for graph labels and weights.

AnimateGraph CircularEmbedding
DilateVertices GraphOptions
Highlight RadialEmbedding
RandomVertices RankGraph
RankedEmbedding RootedEmbedding
RotateVertices ShakeGraph
ShowGraph ShowGraphArray
ShowLabeledGraph SpringEmbedding
TranslateVertices Vertices

Combinatorica functions for drawing graphs.

Generating Graphs

Many graphs consistently prove interesting, in the sense that they are models of important binary relations or have unique graph theoretic properties. Often, these graphs can be parameterized, such as the complete graph on n vertices K_n, giving a concise notation for expressing an infinite class of graphs. Start off with several operations that act on graphs to give different graphs and which, together with parameterized graphs, give the means to construct essentially any interesting graph.

The union of two connected graphs has two connected components.


ShowGraph[ GraphUnion[ CompleteGraph[3],
                        CompleteGraph[5,5] ] ];


Graph products can be very interesting. The embedding of a product has been designed to show off its structure, and is formed by shrinking the first graph and translating it to the position of each vertex in the second graph.


ShowGraph[ GraphProduct[ CompleteGraph[3],
        CompleteGraph[5] ] ];


The line graph L (G) of a graph G has a vertex of L (G) associated with each edge of G, and an edge of L (G) if, and only if, the two edges of G share a common vertex.


ShowGraph[ LineGraph[CompleteGraph[5]] ];


Circulants are graphs whose adjacency matrix can be constructed by rotating a vector n times, and include complete graphs and cycles as special cases. Even random circulant graphs have an interesting, regular structure.


ShowGraph[ CirculantGraph[21,


Some graph generators create directed graphs with self-loops, such as this de Bruijn or shift register graph encoding all length-5 substrings of a binary alphabet.


ShowGraph[ DeBruijnGraph[2,5] ];


Hypercubes of dimension d are the graph product of cubes of dimension d - 1 and the complete graph K_2. Here, a Hamiltonian cycle of the hypercube is highlighted. Colored highlighting and graph animations are also provided in the package.


ShowGraph[ Highlight[Hypercube[4],
{Partition[HamiltonianCycle[Hypercube[4]], 2, 1]}] ];


Several of the built-in graph construction functions do not have parameters and only construct a single interesting graph. FiniteGraphs collects them together in one list for convenient reference. ShowGraphArray permits the display of multiple graphs in one window.




BooleanAlgebra ButterflyGraph
CageGraph CartesianProduct
ChvatalGraph CirculantGraph
CodeToLabeledTree CompleteBinaryTree
CompleteGraph CompleteKPartiteGraph
CompleteKaryTree CoxeterGraph
CubeConnectedCycle CubicalGraph
Cycle DeBruijnGraph
DodecahedralGraph EmptyGraph
ExactRandomGraph ExpandGraph
FiniteGraphs FolkmanGraph
FranklinGraph FruchtGraph
FunctionalGraph GeneralizedPetersenGraph
GraphComplement GraphDifference
GraphIntersection GraphJoin
GraphPower GraphProduct
GraphSum GraphUnion
GridGraph GrotztschGraph
Harary HasseDiagram
HeawoodGraph HerschelGraph
Hypercube IcosahedralGraph
IntervalGraph KnightsTourGraph
LabeledTreeToCode LeviGraph
LineGraph ListGraphs
MakeGraph McGeeGraph
MeredithGraph MycielskiGraph
NoPerfectMatchingGraph NonLineGraphs
OctahedralGraph OddGraph
OrientGraph Path
PermutationGraph PetersenGraph
RandomGraph RandomTree
RealizeDegreeSequence RegularGraph
RobertsonGraph ShuffleExchangeGraph
SmallestCyclicGroupGraph Star
TetrahedralGraph ThomassenGraph
TransitiveClosure TransitiveReduction
Turan TutteGraph
Uniquely3ColorableGraph UnitransitiveGraph
VertexConnectivityGraph WaltherGraph

Combinatorica functions for graph constructors.

Properties of Graphs

Graph theory is the study of properties or invariants of graphs. Among the properties of interest are such things as connectivity, cycle structure, and chromatic number. Here we demonstrate how to compute several different graph invariants.

An undirected graph is connected if a path exists between any pair of vertices. Deleting an edge from a connected graph can disconnect it. Such an edge is called a bridge.


ConnectedQ[ DeleteEdge[ Star[10], {1,10} ] ]



GraphUnion can be used to create disconnected graphs.


ConnectedComponents[ GraphUnion[CompleteGraph[3],
        CompleteGraph[4]] ]


{{1, 2, 3}, {4, 5, 6, 7}}

An orientation of an undirected graph G is an assignment of exactly one direction to each of the edges of G. Note that arrows denoting the direction of each edge are automatically drawn in displaying directed graphs.


ShowGraph[ OrientGraph[Wheel[10]] ];


An articulation vertex of a graph G is a vertex whose deletion disconnects G. Any graph with no articulation vertices is said to be biconnected. A graph with a vertex of degree 1 cannot be biconnected, since deleting the other vertex that defines its only edge disconnects the graph.


        RealizeDegreeSequence[{4,4,3,3,3,2,1}] ]


{{2, 7}, {1, 2, 3, 4, 5, 6}}

The only articulation vertex of a star is its center, even though its deletion leaves n - 1 connected components. Deleting a leaf leaves a connected tree.


ArticulationVertices[ Star[10] ]



Every edge in a tree is a bridge.


Bridges[ RandomTree[10] ]


{{3, 5}, {3, 9}, {7, 9}, {6, 7}, {1, 6}, {2, 8}, {2, 10}, {4, 10}, {1, 10}}

A graph is said to be k-connected if there does not exist a set of k - 1 vertices whose removal disconnects the graph. The wheel is the basic triconnected graph.





A graph is k-edge-connected if there does not exist a set of k - 1 edges whose removal disconnects the graph. The edge connectivity of a graph is at most the minimum degree δ, since deleting those edges disconnects the graph. Complete bipartite graphs realize this bound.





These two complete bipartite graphs are isomorphic, since the order of the two stages is simply reversed. Here, all isomorphisms are returned.


Isomorphism[CompleteGraph[3,2], CompleteGraph[2,3], All]


{{3, 4, 5, 1, 2}, {3, 4, 5, 2, 1}, {3, 5, 4, 1, 2}, {3, 5, 4, 2, 1}, {4, 3, 5, 1, 2}, {4, 3, 5 ...  5, 3, 1, 2}, {4, 5, 3, 2, 1}, {5, 3, 4, 1, 2}, {5, 3, 4, 2, 1}, {5, 4, 3, 1, 2}, {5, 4, 3, 2, 1}}

A graph is self-complementary if it is isomorphic to its complement. The smallest nontrivial self-complementary graphs are the path on four vertices and the cycle on five.


SelfComplementaryQ[ Cycle[5] ] &&
        SelfComplementaryQ[ Path[4] ]



A directed graph in which half the possible edges exist is almost certain to contain a cycle. Directed acyclic graphs are often called DAGs.


        RandomGraph[100, 0.5, Type -> Directed]



The girth of a graph is the length of its shortest cycle. A cage is the smallest possible regular graph (here degree 3) that has a prescribed girth.


Girth[ CageGraph[3, 6] ]



An Eulerian cycle is a complete tour of all the edges of a graph. An Eulerian cycle of a bipartite graph bounces back and forth between the stages.


EulerianCycle[ CompleteGraph[4,4] ]


{7, 2, 8, 1, 5, 4, 6, 3, 7, 4, 8, 3, 5, 2, 6, 1, 7}

AcyclicQ AntiSymmetricQ
BiconnectedQ BipartiteQ
CliqueQ CompleteQ
ConnectedQ EmptyQ
EquivalenceRelationQ EulerianQ
GraphicQ HamiltonianQ
IdenticalQ IndependentSetQ
IsomorphicQ IsomorphismQ
MultipleEdgesQ PartialOrderQ
PerfectQ PlanarQ
PseudographQ ReflexiveQ
RegularQ SelfComplementaryQ
SelfLoopsQ SimpleQ
SymmetricQ TransitiveQ
TreeIsomorphismQ TreeQ
TriangleInequalityQ UndirectedQ
UnweightedQ VertexCoverQ

Combinatorica functions for graph predicates.

A Hamiltonian cycle of a graph G is a cycle that visits every vertex in G exactly once, as opposed to an Eulerian cycle that visits each edge exactly once. K_ (n, n) for n>1 are the only Hamiltonian complete bipartite graphs.


HamiltonianCycle[CompleteGraph[3,3], All]


{{1, 4, 2, 5, 3, 6, 1}, {1, 4, 2, 6, 3, 5, 1}, {1, 4, 3, 5, 2, 6, 1}, {1, 4, 3, 6, 2, 5, 1}, { ... 4, 1}, {1, 6, 2, 4, 3, 5, 1}, {1, 6, 2, 5, 3, 4, 1}, {1, 6, 3, 4, 2, 5, 1}, {1, 6, 3, 5, 2, 4, 1}}

The divisibility relation between integers is reflexive, since each integer divides itself, and anti-symmetric, since x cannot divide y if x>y. Finally, it is transitive, as x∖y implies y = c x for some integer c, so y∖z implies x∖z.


        g = MakeGraph[Range[8],(Mod[#1,#2]==0)&],
        VertexNumber -> True


Since the divisibility relation is reflexive, transitive, and anti-symmetric, it is a partial order.





A graph G is transitive if any three vertices x, y, z, such that edges {x, y}, {y, z} ∈G, imply {x, z} ∈G. The transitive reduction of a graph G is the smallest graph R (G) such that C (G) = C (R (G)). The transitive reduction eliminates all implied edges in the divisibility relation, such as 4∖8, 1∖4, 1∖6, and 1∖8.


ShowGraph[TransitiveReduction[g], VertexNumber -> True];


The Hasse diagram clearly shows the lattice structure of the Boolean algebra, the partial order defined by inclusion on the set of subsets.


  ((Intersection[#2,#1]===#1)&&(#1 != #2))&]]


A topological sort is a permutation p of the vertices of a graph such that an edge {i, j} implies i appears before j in p. A complete directed acyclic graph defines a total order, so there is only one possible output from TopologicalSort.


        MakeGraph[Range[10],(#1 > #2)&] ]


{10, 9, 8, 7, 6, 5, 4, 3, 2, 1}

Any labeled graph G can be colored in a certain number of ways with exactly k colors 1, …, k. This number is determined by the chromatic polynomial of the graph.


        GraphUnion[CompleteGraph[2,2], Cycle[3]], z ]


-6 z^2 + 21 z^3 - 29 z^4 + 20 z^5 - 7 z^6 + z^7

ArticulationVertices Automorphisms
BiconnectedComponents Bridges
ChromaticNumber ChromaticPolynomial
ConnectedComponents DegreeSequence
Degrees DegreesOf2Neighborhood
Diameter Distances
Eccentricity EdgeChromaticNumber
EdgeColoring EdgeConnectivity
Equivalences EulerianCycle
Girth GraphCenter
GraphPolynomial HamiltonianCycle
InDegree Isomorphism
M MaximalMatching
MaximumAntichain MaximumClique
MaximumIndependentSet MaximumSpanningTree
MinimumChainPartition MinimumSpanningTree
MinimumVertexColoring MinimumVertexCover
OutDegree Radius
Spectrum StronglyConnectedComponents
TreeToCertificate V
VertexColoring VertexConnectivity
VertexCover WeaklyConnectedComponents

Combinatorica functions for graph invariants.

Algorithmic Graph Theory

Finally, there are several invariants of graphs that are of particular interest because of the algorithms that compute them.

The shortest-path spanning tree of a grid graph is defined in terms of Manhattan distance, where the distance between points with coordinates (x, y) and (u, v) is x - u + y - v.


    GridGraph[5,5],1] ];


In an unweighted graph, there can be many different shortest paths between any pair of vertices. This path between two opposing corners goes all the way to the right, then all the way to the top.




{1, 2, 3, 4, 5, 10, 15, 20, 25}

A minimum spanning tree of a weighted graph is a set of n - 1 edges of minimum total weight that form a spanning tree of the graph. Any spanning tree is a minimum spanning tree when the graphs are unweighted.


ShowGraph[ MinimumSpanningTree[ CompleteGraph[6,6,6] ] ];


The number of spanning trees of a complete graph is n^(n - 2), as was proved by Cayley.





The maximum flow through an unweighted complete bipartite graph G is the minimum degree δ (G).


NetworkFlow[ CompleteGraph[4,4], 1, 8 ]



A matching, in a graph G, is a set of edges of G such that no two of them share a vertex in common. A perfect matching of an even cycle consists of alternating edges in the cycle.


BipartiteMatching[ Cycle[8] ]


{{1, 2}, {3, 4}, {5, 6}, {7, 8}}

Any maximal matching of a K_n is a maximum matching, and perfect if n is even.




{{1, 2}, {3, 4}, {5, 6}, {7, 8}}

AllPairsShortestPath AlternatingPaths
ApproximateVertexCover BellmanFord
BipartiteMatching BipartiteMatchingAndCover
BreadthFirstTraversal BrelazColoring
CostOfPath DepthFirstTraversal
Dijkstra ExtractCycles
FindCycle GreedyVertexCover
Neighborhood NetworkFlow
ParentsToPaths ResidualFlowGraph
ShortestPath ShortestPathSpanningTree
StableMarriage TopologicalSort
TravelingSalesman TravelingSalesmanBounds

Combinatorica functions for graph algorithms.

Created by Mathematica  (December 5, 2003)