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
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)
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.