mgf 1107 – explorations in mathematics lecture 13

Having now created some graphs from physical settings, we are now in a
position to introduce some additional vocabulary.
Definition: We say that two vertices are adjacent if they are joined by an
Definition: The degree of a vertex is the number of edges connected to
Definitions: A vertex with an even number of edges connected to it is
called an even vertex and a vertex with an odd number of edges
connected to it is called an odd vertex.
Definitions: A path is a sequence of adjacent vertices and the edges
connecting them. The number of edges involved in a path is called the
length. If a path starts and ends at the same point, we call it a circuit.
Note: Every circuit is a path, but not every path is a circuit, since a
circuit needs to start and end at the same vertex.
It is important to recognize the difference between a graph and a path. A
graph is a set of vertices and edges connecting them. A path can be
thought of as movement from vertex to vertex. To represent a path we
use a sequence of vertices separated by commas.
Ex. (5.A.11)
Ex. (5.A.13)
Definitions: A graph is connected if for any two vertices in the graph,
there is a path that connects them. So far all the graphs we have studied
(except for the bowling league example) have been connected. If a graph
is not connected we say it is disconnected.
Definition: A bridge is an edge that if removed from a connected graph
would make it disconnected.
Ex. (5.A.15a)
Note In the Königsberg bridge problem, none of the edges that represent
real bridges are bridges in the graph-theory sense!
Euler Paths and Euler Circuits
Although we looked at the Königsberg bridge problem in the previous
section, we didn’t actually solve it! Before we can do this we need to
introduce two more definitions.
Definition: An Euler path is a path that passes through each edge of a
graph exactly one time.
Definition: An Euler circuit is a circuit that passes through each edge of
a graph exactly one time.
The difference between the two is that an Euler circuit must start and
end at the same vertex.
Ex. For the graphs shown below, determine whether an Euler path or
Euler circuit exists.