MA2005 – Graphs and Networks
2.
Graphs 2
Walks, Eulerian Graphs, Hamiltonian Graphs, Disconnecting Sets
and Trees
Aims
-
To use adjacency matrix to determine the number of walks between vertices of a
graph.
To introduce and discuss Eulerian graphs.
To introduce the idea of a Hamiltonian graph and to present some simple criteria
for determining whether a graph is Hamiltonian.
To introduce concepts relating to how a connected graph can be disconnected.
To discuss briefly some properties of trees.
2.1 Walks
A walk in a graph G is a finite sequence of edges of the form
v0 v1 , v1v2 , v2 v3 ,, vm1vm
(also written v0 v1v2 v3 vm ).
A walk must have at least one edge. The number of edges in the walk is called its length.
e.g. uzyx is a walk of length 3, and uzuzyx is a walk of length 5.
u
z
v
w
y
x
A walk, v0 v1v2 v3 vm , is closed if v0  v m , and open if v0  v m . A closed walk in which all
vertices are distinct (except v0  v m ) is called a cycle (or circuit).
e.g. in the above graph, uvzu is a cycle of length 3 (a triangle).
Counting walks
Let A  aij  be the adjacency matrix of a graph G with vertex set {v1 , v2 ,, vn } . The
(i, j ) th element of A 2 is

n
k 1
aik akj and this is the number of walks of length 2 from vi to
vj .
Example. If G is
v1
v2
v3
v4
1 1 0

0 1 1
1 0 1

1 1 0 
2

1
2
A 
1

2

then
0

1
A
1

0

and
1 1 2

3 2 1
.
2 3 1

1 1 2 
Hence, for example, the number of walks of length 2 from v2 to v3 is 2, and the number of
walks of length 2 from v2 to v2 is 3.
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MA2005 – Graphs and Networks
Graphs 2
Generally, for any positive integer r, the number of walks of length r from vi to vj is given by
the (i, j ) th element of A r .
Trails
A trail is a walk in which no edge has been traversed more than once (in either direction) but
repeated vertices are allowed. It is closed if the last vertex is the same as the first, and open
otherwise.
Example. In the graph, K 5 , below,
- acbdce is an open trail
- acbdcea is a closed trail
- acbdcbe is not a trail, since edge cb has been traversed twice, it is a walk.
a
e
b
d
c
Paths
A path in a graph is a walk with no repeated edges or vertices (except, possibly, the first and
last vertex, in which case we have a closed path, or a cycle). e.g. abce and cdac in above.
2.2 Eulerian and Semi-Eulerian Graphs
Applications: post delivery, rubbish collection, etc.
A connected graph G is Eulerian if there is a closed trail containing every edge of G. We call
such a trail an Eulerian trail.
Examples. (i)
(ii)
Theorem (Euler 1736). A connected graph is Eulerian if and only if every vertex has even
degree.
To find an Eulerian trail in a given graph, start in an arbitrary vertex and traverse along the
edges, ensuring all the edges are traversed before returning to the starting vertex.
If G is not Eulerian, but there is an open trail containing every edge of G, then G is semiEulerian.
Example.
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Graphs 2
Theorem. A connected graph is semi-Eulerian if and only if precisely two of its vertices
have odd degree.
To obtain a semi-Eulerian trail in a given graph, you must start at one of the odd degree
vertices and end in the other odd degree vertex.
The following is an example of a graph that is neither Eulerian nor semi-Eulerian.
2.3 Hamiltonian and semi-Hamiltonian Graphs
A graph G is Hamiltonian if there is a cycle that passes through every vertex of G. Such a
cycle is a Hamiltonian cycle.
If G is not Hamiltonian, but there is an open path which includes every vertex of G, then G is
semi- Hamiltonian, and such a path is a Hamiltonian path.
Examples.
Hamiltonian
semi- Hamiltonian
neither Hamiltonian
nor semi- Hamiltonian
Theorem (Ore 1960).
Let G be a simple graph with n  3 vertices. If deg( u )  deg( v)  n for each pair of nonadjacent vertices u and v, then G is Hamiltonian.
Corollary (Dirac 1952).
Let G be a simple graph with n  3 vertices. If deg( v)  n / 2 for every vertex v, then G is
Hamiltonian.
Note that this is a necessary condition, but not sufficient.
Example. The Petersen graph in non-Hamiltonian but K n , (n  3) graphs are Hamiltonian.
Bipartite graphs can only be Hamiltonian if sets A and B of vertices (as previously defined)
have the same number of vertices. It follows that if the total number of vertices in a bipartite
graph is odd then it cannot be Hamiltonian.
There is no known general criterion for testing whether a graph is semi-Hamiltonian.
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MA2005 – Graphs and Networks
Graphs 2
Exercises 2.1
1. Write down the adjacency matrix A for K 5 . By working out A3 determine
(i)
the number of walks of length 3 from a vertex u to a distinct vertex v;
(ii)
the number of triangles (i.e. cycles of length 3).
2. Determine whether the following graphs are Eulerian, semi-Eulerian or neither. For
those that are Eulerian or semi-Eulerian, find a suitable trail.
v2
(i)
(ii)
v1
v1
v5
v6
v2
v7
v8
v3
v7
(iii)
v1
v9
v8
v4
(iv)
v6
v2
3.
v5
v1
v6
v2
v1
v9
v7
v8
v3
v6
(v)
v9
v5
v4
v4
v3
v2
v9
v5
v7
v4
v6
v8
v3
v5
v7
v4
v8
v3
In the Petersen graph,
(i) Find walks of lengths 5, 6 and 8.
(ii) Find cycles of lengths 5, 6, 8 and 9.
(iii) Can you find a cycle of length 10?
(iv) What is the size of the smallest cycle?
4. Bridges of Konigsberg. Euler lived in the town of Konigsberg that had rivers passing
through it (as shown below) and with bridges connecting four parts of the town. He
considered the following problem: Is it possible to cross each of the seven bridges exactly
once and return to your starting point? Hint: Let the four parts A, B, C and D be vertices of a
suitable non-simple graph.
C
A
D
B
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MA2005 – Graphs and Networks
Graphs 2
5. Determine whether the following graphs are Hamiltonian, semi- Hamiltonian or neither.
Find a Hamiltonian cycle or path if one exists.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
2.4 Disconnecting Sets (this involves edges)
Recall that a graph is connected if there is a path joining each pair of vertices.
Let G be a connected graph. By a disconnecting set we mean a set of edges whose
deletion results in a disconnected graph.
Example. In graph G1, e2 , e3 , e4  and e1 ,e2  are disconnecting sets but e3 , e5  is not.
e2
e7
G1
e1
e3
e5
e6
e4
If a disconnecting set has only one edge e (as in G2 below), then e is called a bridge (or
cut-edge).
e
G2
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MA2005 – Graphs and Networks
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The edge connectivity of G is the size of the smallest disconnecting set of G, in other
words, it is the smallest number of edges whose deletion disconnects G. It is denoted by
 G  .
In the graph G1 above since e1 ,e2  is a disconnecting set of size 2, then  G1   2.
Result
Let  G  denote the minimum degree of G. Then  G   (G) .
2.5 Separating Sets (this involves vertices)
A separating set is a set of vertices whose removal disconnects G. When removing
vertices, we must also remove any incident edges.
Example. In graph G3, u, v is a separating set, but u is not.
u
G3
v
If a separating set contains only one vertex w, then w is called a cut-vertex.
The vertex connectivity of a connected graph G is the size of the smallest separating set.
In other words, it is the smallest number of vertices whose removal disconnects G. It is
denoted by  G  .
Theorem. In a connected graph G, we have  G   (G)   (G) , for G  K n .
Examples.
  2,   1 .
K 4 :   3,  does not exist for K n .
K 3,3 :   3,   3 .
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MA2005 – Graphs and Networks
Graphs 2
2.6 Trees
A tree is a connected graph without cycles. If G is a tree then the following holds.
(i)
(ii)
(iii)
(iv)
(v)
G has n-1 edges, where n is the number of vertices of G;
Any two vertices in G are connected by exactly one path;
Each edge of G is a bridge;
The addition of any new edge to G creates exactly one cycle;
G is bipartite, i.e. all of its vertices can be labelled with either a or b so that
every edge is an ab edge (no aa or bb edges).
Examples.
One application of this type of graph is to represent the arrangement of atoms in a
molecule.
Exercises 2.2
1. A cutset is a disconnecting set, no proper subset of which is a disconnecting set. In
the Petersen graph find cutsets with 3, 4 and 5 edges.
2. Find the vertex connectivity and the edge connectivity for each of the following
graphs, giving reasons for your answers.
(i)
(ii)
(iii)
(iv)
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