CMSC203 Discrete Structures, Sections 0201, Spring 2008
The material for this class is drawn from several sources, sometimes with
conflicting terminology. Here are the "official" definitions for some of
the mathematical terms we are using in this class.
In graph theory, two vertices of a graph
are adjacent if there is an edge between the two vertices.
In graph theory, the degree of a vertex u in an undirected graph is the number of edges that
are incident on u (i.e., the number of edges
that have u as an endpoint). A self loop (an edge from a vertex
to itself) adds 2 to the degree of that vertex.
Pronounced "Oiler" not "You-ler". Born 1707, died 1783.
One of the greatest mathematicians of all time.
An Euler graph (also called an Eulerian graph) is an undirected graph which has a circuit
that uses every edge in the graph exactly once.
A graph G = (V, E) consists of a non-empty set of vertices V and a
set edges E. Each edge (u,v) ∈ E is a pair of vertices. That
is, E ⊆ V × V. In an undirected graph the order
of the vertices in an edge does not matter, so (u,v) = (v,u). [This
is an abuse of the ordered pair notation but is commonly done in
computer science literature.] In a directed graph the order
of the vertices does matter and (u,v) ≠ (v,u). Graphs are drawn as
circles and lines. For pictures, follow the links below.
We will use the term "vertex" and "node" interchangeably.
Defined this way, there can be at most one edge between two vertices
since E is a set and can have at most one "copy" of (u,v). Some people call
such graphs simple graphs, we will just say "graphs". Graphs that
are allowed to have multiple edges between the same pair of vertices will
be called multigraphs. In a multigraph, the set of edges E will have
to be a multiset.
Since the set of vertices V is required to be a non-empty set, we do not
allow a graph with zero vertices. However, a graph with an empty set of
edges is allowed.
An edge, either directed or undirected, from a vertex to itself is called a
self-loop. We allow graphs to have self-loops unless otherwise
A coloring of an undirected graph G = (V,E)
is an assignment of colors to the vertices of G such that
for every edge (u,v) ∈ E, the vertices u and v are assigned different
This type of coloring is also called a vertex coloring. Sometimes
people consider coloring the edges of a graph. For this class, we'll stick
to coloring vertices.
In graph theory, an edge (u,v) is said to be incident on the
vertices u and v.
This is one of those terms in mathematics that is much easier to
understand intuitively than to define formally. Draw a graph. Put your finger on a vertex. Follow an
edge out of that vertex. Keep following edges until you stop. Your
finger has traced out a path in the graph.
Formally, a path is a sequence of edges
in a graph such that for all i = 1, ..., n-1, vi =
ui+1. I.e., the end point of an edge in the path must
the starting point of the next edge in the path.
We say that n is the length of the path.
If u1 = vn (i.e., the path ends at its
starting point), then the path is called a circuit.
The term "simple" is often used to modify path and circuit. Some
people use "simple path" to indicate paths that do not repeat
vertices. Others use "simple path" to indicate paths that do not
repeat edges. This is too confusing. We will simply say a path
that does not repeat a vertex or a path that does not repeat
an edge and refrain from using the word "simple".
A planar graph is a graph that can be "drawn" on
piece of paper without any of the edges crossing. This is an informal
definition, but will do for this class. A formal definition would require
us to talk about embedding edges in a plane which is more complicated than
we need. The famous Four-Color Theorem states that every planar graph can
be colored using four colors.
7 Feb 2008 09:03:43 EST
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