CME 305: Discrete Mathematics and Algorithms
Instructor: Professor Amin Saberi ([email protected])
January 12, 2009
Lecture 3: Eulerian Circuits
This lecture is about one of the ﬁrst known applications of graph theory, Eulerian circuits. We give some
basic deﬁnitions, prove Euler’s result, and then illustrate an application from bioinformatics.
1
Eulerian Circuits
Deﬁnition:
A closed walk (circuit) on graph
G
(
V,E
) is an
Eulerian circuit
if it traverses each edge in
E
exactly once. We call a graph
Eulerian
if it has an Eulerian circuit.
The problem of ﬁnding Eulerian circuits is perhaps the oldest problem in graph theory. It was originated by
Euler in the 18th century; the problem was whether one could take a walk in K¨onigsberg and cross each of
the four bridges exactly once. Motivated by this, Euler was able to prove the following theorem:
Theorem 1
(Euler, 1736)
Graph
G
(
V,E
)
is Eulerian iﬀ
G
is connected (except for possible isolated vertices) and the degree of every
vertex in
G
is even.
Proof:
“
⇒
”: assume
G
has an Eulerian circuit. Clearly,
G
must be connected, otherwise we will be unable to
traverse all the edges in a closed walk. Suppose that an Eulerian circuit starts at
v
1
; note that every time
the walk enters vertex
v
6
=
v
1
, it must leave it by traversing a new edge. Hence, every time that the walk
visits
v
it traverses two edges of
v
. Thus
d
v
is even. The same is true for
v
1
except for the ﬁrst step of the
walk that leaves
v
1
and the last step that it enters
v
1
. Again, we can pair the ﬁrst and last edges in the
circuit to conclude that
d
v
1
is also even.
“
⇐
”: we prove this by strong induction on the number of nodes. For
n
= 1, the statement is trivial. Assume
G
has
k
+ 1 vertices, it is connected, and all the nodes have even degrees. Pick any vertex
v
∈
V
and start
walking through the edges of the graph, traversing each edge at most once. Given that all the degrees of
the nodes in
G
are even, we can only get stuck at
v
. Remove the circuit of visited edges,
c
1
, from
G
; Call
this new graph
G
0
. Node
v
is an isolated vertex in
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 Winter '09
 Graph Theory, Eulerian path, Hamiltonian path, Eulerian Circuits, Eulerian Circuit

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