# Definitions

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.

Index: [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]

## A

In graph theory, two vertices of a
graph are adjacent if there is an edge between the two vertices.

## C

Circuit: See path.

## D

#### Degree

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.

Directed Graph: See graph.

## E

#### Euler, Leonhard

Pronounced "Oiler" not "You-ler". Born 1707, died 1783. One of the greatest mathematicians of all time.

#### Euler Graph

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.

## G

#### Graph

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 specified.

#### Graph Coloring

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 colors.

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.

## I

#### Incident

In graph theory, an edge (u,v) is said to be incident on the vertices u and v.

## M

Multigraph: See graph.

## P

#### Path

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 (u1,v1), (u2,v2), (u3,v3), ..., (un,vn) 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".

#### Planar Graph

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.

## U

Undirected graph: See graph.