Graph Theory Lecture Notes 1

Example: Cocktail Party Graph

At any party of 6 people there must be a set of at least three people who are mutual friends or a set of three people who are mutual strangers.

We can diagram this situation by representing the people as dots and joining two dots with a red line if they are friends and a blue line if they are strangers. The statement can then be rephrased as; in this diagram (no matter how the red and blue lines are drawn) there must be either a red triangle or a blue triangle. Here is an example of such a diagram:

At any dot there are 5 lines. At least three must have the same color, for definiteness, let's say that at a certain dot there are three red lines. If any two of the other endpoints of these lines are joined by a red line, then there will exist a red triangle. If none of them are joined by a red line, then all three are joined by a blue line, and so, there exists a blue triangle.

Example: Finite State Autonoma

We wish to build a simple machine (or circuit) which recognizes a specific pattern in a bit string and requires no memory (storage). The bit string is read one bit at a time. The following finite state autonoma recognizes bit strings that end in a 1. It consists of three states (start, accept and reject). Start in the start state and for each bit that is read, follow the arrow with that bit as a label to the next state. When the string is completely read, the state you are in tells you whether the string ended in a 1 (accept) or a 0 (reject).

Def: Graph, Digraph

Unless otherwise specified, our graphs (and digraphs) will have finite vertex sets, self-loops and multiple edges (or arcs) are allowed. When no self-loops and multiple edges are present, we refer to the graph as a simple graph.

Def: Adjacent Vertices (Edges) , Degree (valence) of a vertex, Degree sequence of a graph

Theorem 1.1.2 (Euler): The sum of the degrees of the vertices of a graph is twice the number of edges.


Def: Regular Graphs

Here are some some famous little simple graphs that I have known:

K4 : The complete graph on 4 vertices. The K stands for Kuratowski, a pioneer in the area of graph theory. The term complete refers to the fact that all the possible edges are present. In general, the complete graph on n vertices is denoted Kn, and is an (n-1)-regular simple graph.
K3,3 : A complete bipartite graph. (The K still stands for Kuratowski.) This is also known as the 3-utilities graph, or the water-light-gas graph. Bipartite refers to the fact that the vertices can be grouped into two sets, with no edges existing between vertices in the same set. The term complete here means that all possible edges between the two sets exist. (In general, the two sets need not have the same size). The common name for this graph comes from a famous old problem (involving houses and wells) which has been updated. The problem now is stated as: there are three houses and three utility company supply terminals (the water company, the gas company and the electric company). Each utility needs to run a supply line from the terminal to each of the houses. Can these supply lines be situated so that no two of them cross each other (except, of course, at the houses). This problem (whose answer is no), leads to the interesting concept of planarity of graphs.
C4 : The cyclic graph on 4 vertices. In general Cn refers to the cyclic graph on n vertices and is a 2-regular simple graph. Cn is bipartite when n is even.
Petersen's Graph : This graph on 10 vertices and 15 edges is very famous because it tends to be a counter-example to many generalizations of ideas that work for smaller graphs. As a rule of thumb, check any conjecture on the Petersen graph before trying to prove it.
Hypercube Graphs : A hypercube graph Qn is the n-regular graph whose vertex set is the set of bitstrings of length n, and such that there is an edge between two vertices iff they differ in exactly one bit. The diagram above pictures Q4.
Circular Ladder Graphs: The circular ladder graph CLn consists of two concentric n-cycles in which each of the n corresponding vertices is joined by an edge. It is a 3-regular simple graph. The diagram above pictures CL8.