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Some Links:

- Leonhard Euler Biography on Wikipedia
- Leonhard Euler Biography from University of St Andrews, Scotland
- Leonhard Euler on the Mathematics Genealogy Project

Some Links:

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
specified.

Some Links:

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.

Some links:

Formally, a **path** is a sequence of edges
(u_{1},v_{1}),
(u_{2},v_{2}),
(u_{3},v_{3}),
...,
(u_{n},v_{n})
in a graph such that for all i = 1, ..., n-1, v_{i} =
u_{i+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 u_{1} = v_{n} (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".

Some links:

- planar graphs on MathWorld
- planar graphs on Wikipedia
- the Four-Color Theorem on MathWorld
- the Four-Color Theorem on Wikipedia

Last Modified: 7 Feb 2008 09:03:43 EST by Richard Chang to Spring 2008 CMSC 203 Section Homepage