Graph Theory

Graph Theory was developed by Leonhard Euler to solve the 7 bridges of Konigsberg problem.

A graph $G$ is a collection of vertices, $V$, and edges, $E$.

So we say $G = <V,E>$

Here is an example of a graph:

Incident - Edges that touch a vertex. Ex. $x$ is incident to $a$ and $b$.

Adjacent - Vertices that are connected. Ex. $a$ is adjacent to $b$; $b$ is adjacent to $a,c,e$

Isolated - Vertices that are not connected. Ex. $d$ is isolated.

The are two types of graphs, undirected and directed. In a directed graph, order matters.

There are two types of “walks”. X-Y walks and closed walks.

def: the number of edges incident to a node is called the degree of a node.

def: A graph is simple if it has no loops of multiple edges.

Graph Coloring Problem

Given graph $G$ and $k$ colors assign a color to each node so that the adjacent nodes get different colors.

Def. Minimum value of $k$ for which such a coloring exists is the chromatic value. $χ(G)$

Basic graph coloring algorithm for $G=(V,E)$

  1. Order the nodes $V_1, V_2 .. V_n$
  2. Order the colors $C_1, C_2 .. C_k$
  3. For $i = 1,2 \ldots n$ assign lowest legal color for $V1$
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