Undirected graphs and networks

170 General Mathematics
WORKED
Example
5
Eulerian paths and circuits
1 For which of the following graphs is it possible to draw an:
a Eulerian path b Eulerian circuit?
i A B
ii A B
iii
2 Copy and complete this table for the graphs in question 1.
Vertex
A
B
C
D
E
F
Number
of odd
vertices
Eulerian
path
(Yes/No)
Eulerian
circuit
(Yes/No)
21C
D C
D C
D C
iv A B v A D
vi A C
D C
Degree of
vertex in
graph
i
Degree of
vertex in
graph
ii
Degree of
vertex in
graph
iii
Note: The shaded regions indicate that the particular vertex does not exist for the
given graph.
3 Using the results from question 2, copy and complete these rules to check if it is
possible to draw an Eulerian path or an Eulerian circuit.
a An Eulerian path is possible if the number of odd degree vertices is or .
b An Eulerian circuit is possible if the vertices all have degrees.
4 For the following networks:
i state the degree of each vertex
ii specify whether an Eulerian path is possible
E
B C
Degree of
vertex in
graph
iv
A
Degree of
vertex in
graph
v
E
F E
B
B D
Degree of
vertex in
graph
vi