Hamilton path and euler path

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what is Hamilton path and Euler path? History of Euler path and Hamilton path Vertex(node) and edge Hamilton path and Hamilton circuit Euler path and Euler circuit Degree of vertex and comparison of Euler and Hamilton path Solving a problem

Things we are going to discuss
 History of Euler path and Hamilton path
 Vertex(node) and edge
 Hamilton path and Hamilton circuit
 Euler path and Euler circuit
 Degree of vertex and comparison of Euler and Hamilton path
 Solving a problem

History of Euler Path
Euler path was first discussed by
Leonhard Euler while solving the famous
Seven Bridges of Konigsberg problem
in 1736. The problem can be stated
mathematically like this:
a
b
c
d

History of Hamilton Path
 Hamiltonian paths and cycles
are named after William Rowan
Hamilton who invented the
ICOSIAN game, now also
known as Hamilton's puzzle,
which involves finding a
Hamiltonian cycle in the edge
graph of the dodecahedron

Vertex
Vertex :
Vertex typically means a
corner or a point where
lines meet.
Vertex
Not a Vertex

Edge:
An edge is a line segment that
joins two vertices
Edge
Edge

Hamiltonian Path and circuits
In graph theory, an Hamilton path is a path which visits
every vertex exactly once. Similarly, an Hamilton circuit is a
circuit which starts and ends on the same vertex.

 This has a Hamilton Path and a
Hamilton circuit
 This has a Hamilton Path , but does not
have a Hamilton circuit
 This has no Hamilton Path

Euler path and circuit
In graph theory, an Euler path is a path
which visits every edge exactly once.
Similarly, an Eulerian circuit or Eulerian
cycle is an Eulerian trail which starts and
ends on the same vertex.

This has an Euler Path and an Euler
circuit
This has an Euler Path , but does not
have an Euler circuit
This has no Hamilton Path

Degree of Vertex
The degree of a vertex is the number
of edges of the vertex.

Comparison of Euler and Hamilton circuits
Property Euler Hamilton
Repeated visits to a
given node allowed?
Repeated traversals of a
given edge allowed?
Omitted nodes allowed?
Omitted edges allowed?
YES NO
NO NO
NO NO
NO YES

in the image, is it possible to
construct a path ( a path starting and
ending on the same vertex) which
visits each edge exactly once?
Seven Bridges Problem
A
B
C
D
a
b
c
d

Solve
We know that ,
 An Euler Circuit is possible if every vertex has
even degree.
 An Euler path is possible but Euler Circuit is
possible ,if exactly two vertices have odd
degree.
 No Euler Path is possible if more than two
vertices have odd degree.
a
b
c
d
THERE IS NO EULERS PATH

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Hamilton path and euler path

1. Hamilton Path and Euler Path

2. Things we are going to discuss  History of Euler path and Hamilton path  Vertex(node) and edge  Hamilton path and Hamilton circuit  Euler path and Euler circuit  Degree of vertex and comparison of Euler and Hamilton path  Solving a problem

3. History of Euler Path Euler path was first discussed by Leonhard Euler while solving the famous Seven Bridges of Konigsberg problem in 1736. The problem can be stated mathematically like this: a b c d

4. History of Hamilton Path  Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the ICOSIAN game, now also known as Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron

5. Vertex Vertex : Vertex typically means a corner or a point where lines meet. Vertex Not a Vertex

6. Edge: An edge is a line segment that joins two vertices Edge Edge

7. Hamiltonian Path and circuits In graph theory, an Hamilton path is a path which visits every vertex exactly once. Similarly, an Hamilton circuit is a circuit which starts and ends on the same vertex.

8.  This has a Hamilton Path and a Hamilton circuit  This has a Hamilton Path , but does not have a Hamilton circuit  This has no Hamilton Path

9. Euler path and circuit In graph theory, an Euler path is a path which visits every edge exactly once. Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail which starts and ends on the same vertex.

10. This has an Euler Path and an Euler circuit This has an Euler Path , but does not have an Euler circuit This has no Hamilton Path

11. Degree of Vertex The degree of a vertex is the number of edges of the vertex.

12. Comparison of Euler and Hamilton circuits Property Euler Hamilton Repeated visits to a given node allowed? Repeated traversals of a given edge allowed? Omitted nodes allowed? Omitted edges allowed? YES NO NO NO NO NO NO YES

13. in the image, is it possible to construct a path ( a path starting and ending on the same vertex) which visits each edge exactly once? Seven Bridges Problem A B C D a b c d

14. Solve We know that ,  An Euler Circuit is possible if every vertex has even degree.  An Euler path is possible but Euler Circuit is possible ,if exactly two vertices have odd degree.  No Euler Path is possible if more than two vertices have odd degree. a b c d THERE IS NO EULERS PATH

15. THANK YOU !

Hamilton path and euler path

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Hamilton path and euler path