Graph Theory Lecture Notes 3

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.

Def: Graph, Digraph

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

Def: Labeled graph

Let L(n) be the number of labeled graphs having n vertices. Let L(n,e) be the number of labeled graphs having n vertices and e edges. Then

L(n,e) = C( C(n,2), e)
and
L(n) = L(n,0) + L(n,1) + ... + L(n, C(n,2)) = C (C(n,2), 0) + C( C(n,2), 1) + ... + C(C(n,2), C(n,2)) = 2C(n,2).
For labeled digraphs we get a similar formula M(n) = 2n(n-1).

Def: Isomorphic graphs

The graph isomorphism problem is concerned with determining when two graphs are isomorphic. This is a difficult problem, and in the general case there is no known efficient algorithm for doing it.

It is often easy to show that two graphs are not isomorphic. For instance, if they have different numbers of vertices or edges, or if the degrees of the vertices do not match up. But showing that they are isomorphic requires that a matching can actually be produced.

Problem: Are these graphs isomorphic?

Here are some some famous little 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.
K3,3 : A complete bipartite graph. (The K still stands for Kuratowski.) This is also known as the 3-utilities graph, or as our text prefers 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.
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.