# C

## Cage

An (*n,g*)-cage is a graph which
is *n*-regular and has girth equal
to *g*.

A chain (more commonly called a path )
in a graph is a sequence *{x*_{1}, x_{2}, x_{3}, ... x_{k}} of nodes
such that *{[x*_{1}, x_{2}], [x_{2}, x_{3}], ... [x_{k-1}, x_{k}]} are edges of the graph.
In a non-simple graph (one with repeated edges) it is necessary to impose
the condition that no edge occur in the sequence more than it's repetition
number. In a simple graph, each edge can appear at most once.

## Chromatic Number

The chromatic number (gamma of *G*) of a graph is the smallest number
of colors such that colors can be assigned to all nodes of the graph
without having connected nodes with the same color.

## Clique

A fully connected subgraph of a graph .

## Code

A subset of the vectors from a vector space over a finite
field , these vectors are collectively
called "code words" and the finite field is usually *GF(2)*, although
there are useful codes over *GF(3)* and *GF(4)* as well.

In an error-correcting code, the code words are chosen such that the
"distance" between them is maximized, thus small transmission errors
can be recovered by interpreting the received vector as the nearest
code word.

See Hamming distance

## Combination

Let *r* be a non-negative integer. An *r*-combination of a set is an
unordered selection of *r* elements out of the set. Equivalently,
an *r*-combination is an *r*-subset of the original set.

## Contraction

The graph * G:xy * formed by
replacing distinct nodes * x * and * y * with a single
new node * z * which is connected to any nodes that were connected
to either * x * or * y * is called a contraction of
* G *.

## Convex

A set in R^{d} is convex if it contains the line segment
connecting any two of its points.

## Convex Hull

The intersection of convex regions containing a set in R^{d}.

## Coxeter/Dynkin Diagram

A diagram used to visualize a Coxeter group, it
is a labeled graph with nodes indexed by the generators of a
Coxeter group and *(P*_{i} P_{j}) is an edge
whenever *M*_{ij} > 2
which is labeled with *M*_{ij}.
Where *M*_{ij} is an entry in
the Coxeter matrix corresponding to the given
Coxeter group.

## Coxeter group

A group generated by the elements
*P*_{i} with
*i* in *{1,2, ... n}*, subject to the relations

,

where *M*_{ij} are the elements of a
Coxeter matrix.

## Coxeter matrix

A Coxeter matrix of rank *n* is an *n x n* matrix *M*
with *M*_{ii} = 1 and
*M*_{ij} = M_{ji} > 1 (possibly infinite)
for all *i* and *j* in *{1,2, ... n}*.

## Cross Polytope

A regular polytope, the convex hull of the
points formed by permuting the coordinates of
(1,0,0,,...,0) and (-1,0,0,,...,0).
In three dimensions, this is the octahedron.

## Cycle

In graph theory a cycle is a subset of the edge-set of a graph that form
a chain , the first node of which is also the last.

A -
B -
C -
D -
E -
F -
G -
H -
I -
J -
K -
L -
M -
N -
O -
P -
Q -
R -
S -
T -
U -
V -
W -
X -
Y -
Z

An On-line Dictionary of Combinatorics