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.
You are encouraged to visit the website, www.combinatorica.com, where you will find the latest release of the package, an editor for Combinatorica
graphs, and additional files of interest.
Permutations and Combinations
Permutations and subsets are the most basic combinatorial objects. Combinatorica
provides functions for constructing objects both randomly and deterministically, to rank and unrank them, and to compute invariants on them. Here are 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.
The ranking function illustrates that the built-in function Permutations
uses lexicographic sequencing.
distinct permutations of three elements, within 20 random permutations you are likely to see all of them. Observe that it is unlikely for the first six permutations to all be distinct.
A fixed point of a permutation
is an element in the same position in
as in the inverse of
. Thus, the only fixed point in this permutation is 7.
The identity permutation consists of
singleton cycles or fixed points.
The classic problem in Polya theory is counting how many different ways necklaces can be made out of
beads, when there are
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
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.
The number of inversions in a permutation is equal to that of its inverse.
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.
In a Gray code, each subset differs in exactly one element from its neighbors. Observe that the last eight subsets all contain
, while none of the first eight do.
-subset is a subset with exactly
elements in it. Since the lead element is placed in first, the
-subsets are given in lexicographic order.
Combinatorica functions for permutations.
Combinatorica functions for subsets.
Combinatorica functions for group theory.
Partitions, Compositions, and Young Tableaux
A partition of a positive integer
is a set of
strictly positive integers whose sum is
. A composition of
is a particular arrangement of non-negative integers whose sum is
. A set partition of
elements is a grouping of all the elements into nonempty, nonintersecting subsets. A Young tableau is a structure of integers
where the number of elements in each row is defined by an integer partition of
. 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.
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
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. This constructs a random partition of 100.
Here every composition of 5 into 3 parts is generated exactly once.
Set partitions are different than integer partitions, representing the ways you can partition distinct elements into subsets. They are useful for representing colorings and clusterings.
The list of tableaux of shape
illustrates the amount of freedom available to tableau 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.
By iterating through the different integer partitions as shapes, all tableaux of a particular size can be constructed.
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
computes the number of tableaux on
Each of the 117,123,756,750 tableaux of this shape will be selected with equal likelihood.
A pigeonhole result states that any sequence of
distinct integers must contain either an increasing or a decreasing scattered subsequence of length
Combinatorica functions for integer partitions.
Combinatorica functions for set partitions.
Combinatorica functions for Young tableaux.
Combinatorica functions for counting.
A graph is defined as 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
, each vertex is adjacent to all other vertices. CompleteGraph[n]
constructs the complete graph on
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.
The adjacency matrix of
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.
The standard embedding of
consists of five vertices equally spaced on a circle.
The number of vertices in a graph is termed the order of the graph.
returns the number of edges in a graph.
Edge/vertex colors/styles can be globally modified, giving complete flexibility to change the appearance of a graph.
The colors, styles, labels, and weights of individual vertices and edges can also be changed individually, perhaps to highlight interesting features of the graph.
A star is a tree with one vertex of degree
. Adding any new edge to a star produces a cycle of length 3.
Graphs with multi-edges and self-loops are supported. Here there are two copies of each edge of a star.
The adjacency list representation of a graph consists of
lists: one list for each vertex
, which records the vertices to which
is adjacent. Each vertex in the complete graph is adjacent to all other vertices.
ordered pairs of edges defined by a complete graph of order
An induced subgraph of a graph
is a subset of the vertices of
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.
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.
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.
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.
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 is a radial embedding of a random labeled tree.
By arbitrarily selecting a root, any tree can be represented as a rooted tree.
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
is connected to each of two disconnected vertices. In achieving the minimum energy configuration, these two vertices end up on different sides of
Combinatorica functions for modifying graphs.
Combinatorica functions for graph format translation.
Combinatorica options for graph functions.
Combinatorica functions for graph labels and weights.
Combinatorica functions for drawing 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
, 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.
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.
The line graph
of a graph
has a vertex of
associated with each edge of
, and an edge of
if and only if the two edges of
share a common vertex.
Circulants are graphs whose adjacency matrix can be constructed by rotating a vector
times, and include complete graphs and cycles as special cases. Even random circulant graphs have an interesting, regular structure.
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.
are the graph product of cubes of dimension
and the complete graph
. Here, a Hamiltonian cycle of the hypercube is highlighted. Colored highlighting and graph animations are also provided in the package.
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.
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 is a demonstration of 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.
can be used to create disconnected graphs.
An orientation of an undirected graph
is an assignment of exactly one direction to each of the edges of
. Note that arrows denoting the direction of each edge are automatically drawn in displaying directed graphs.
An articulation vertex of a graph
is a vertex whose deletion disconnects
. 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.
The only articulation vertex of a star is its center, even though its deletion leaves
connected components. Deleting a leaf leaves a connected tree.
Every edge in a tree is a bridge.
A graph is said to be
-connected if there does not exist a set of
vertices whose removal disconnects the graph. The wheel is the basic triconnected graph.
A graph is
-edge-connected if there does not exist a set of
edges whose removal disconnects the graph. The edge connectivity of a graph is at most the minimum degree
, 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.
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.
A directed graph in which half the possible edges exist is almost certain to contain a cycle. Directed acyclic graphs are often called DAGs.
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.
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.
Combinatorica functions for graph predicates.
A Hamiltonian cycle of a graph
is a cycle that visits every vertex in
exactly once, as opposed to an Eulerian cycle that visits each edge exactly once.
are the only Hamiltonian complete bipartite graphs.
The divisibility relation between integers is reflexive, since each integer divides itself, and anti-symmetric, since
. Finally, it is transitive, as
for some integer
Since the divisibility relation is reflexive, transitive, and anti-symmetric, it is a partial order.
is transitive if any three vertices
, such that edges
. The transitive reduction of a graph
is the smallest graph
. The transitive reduction eliminates all implied edges in the divisibility relation, such as
The Hasse diagram clearly shows the lattice structure of the Boolean algebra, the partial order defined by inclusion on the set of subsets.
A topological sort is a permutation
of the vertices of a graph such that an edge
. A complete directed acyclic graph defines a total order, so there is only one possible output from TopologicalSort
Any labeled graph
can be colored in a certain number of ways with exactly
. This number is determined by the chromatic polynomial of the graph.
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
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.
A minimum spanning tree of a weighted graph is a set of
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.
The number of spanning trees of a complete graph is
, as was proved by Cayley.
The maximum flow through an unweighted complete bipartite graph
is the minimum degree
A matching, in a graph
, is a set of edges of
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.
Any maximal matching of a
is a maximum matching, and perfect if
Combinatorica functions for graph algorithms.