TECHNICAL PRESENTATIONTECHNICAL PRESENTATIONONON

HAMILTONIAN HAMILTONIAN CIRCUITCIRCUIT

Presented By :Presented By :ABHISHEK KUMAR SINGH ABHISHEK KUMAR SINGH 33RDRD -07RWSCA001 -07RWSCA001

OutlineOutline Introduction Definition Example Real life applications Discussion

Introduction

Hamiltonian path is a Hamiltonian path is a pathpath in an in an undirected graphundirected graph which visits each which visits each vertexvertex exactly once. A Hamiltonian cycle (or exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian circuit) is a cyclecycle in an in an undirected graphundirected graph which visits each which visits each vertexvertex exactly once and also returns to the exactly once and also returns to the starting vertex. Determining whether such starting vertex. Determining whether such paths and cycles exist in graphs is the paths and cycles exist in graphs is the Hamiltonian path probleHamiltonian path problemm

Paths and CircuitsPaths and Circuits A walk in a graph is an alternating A walk in a graph is an alternating

sequence of adjacent vertices and edges.sequence of adjacent vertices and edges.

A path is a walk that does not contain a A path is a walk that does not contain a repeated edge.repeated edge.

A circuit is a closed walk that does not A circuit is a closed walk that does not contain a repeated edge.contain a repeated edge.

A simple circuit is a circuit which does not A simple circuit is a circuit which does not have a repeated vertex except for the first have a repeated vertex except for the first and last.and last.

A A Hamilton pathHamilton path is a path that goes is a path that goes through each vertex of the graph through each vertex of the graph once and only once.once and only once.

F, A, B, E, C, G, D is a Hamilton path

A A Hamilton circuitHamilton circuit is a circuit that is a circuit that goes through each vertex of the goes through each vertex of the graph once and only once (starting graph once and only once (starting point and ending point is the same) point and ending point is the same)

F, B, E, C, G, D, A, F is a Hamilton circuit

A A complete graphcomplete graph with N vertices is a with N vertices is a graph in which every pair of distinct graph in which every pair of distinct vertices is joined by an edge. Symbol vertices is joined by an edge. Symbol is Kis KNN

KKNN has has N(N-1) / 2N(N-1) / 2 edges edges

The number of Hamilton circuits in KThe number of Hamilton circuits in KNN is is (N-1)!(N-1)! ExampleExample This complete graph has 4 vertices so there areThis complete graph has 4 vertices so there are(4-1)! = 3! = 3(4-1)! = 3! = 3·2 ·1= 6 Hamilton circuit·2 ·1= 6 Hamilton circuit

Let A be the reference point:Let A be the reference point:A, B, C, D, AA, B, C, D, AA, B, D, C, AA, B, D, C, AA, C, B, D, AA, C, B, D, AA, C, D, B, AA, C, D, B, AA, D, B, C, AA, D, B, C, AA, D, C, B, AA, D, C, B, A

A B

CD

Graph of the tetrahedron with a Hamiltonian circuit

Graph of the cube with a Hamiltonian circuit

Example

Real life applications: Anything where you have to visit all locations,

such as :-

pizza delivery

mail delivery

traveling salesman

garbage pickup

reading gas meters

Any Question on???Any Question on???

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