This document discusses the Hamiltonian path problem in graph theory. A Hamiltonian path visits each vertex in a graph exactly once. The Hamiltonian path problem is determining if a Hamiltonian path exists in a given graph. It is computationally difficult to solve and several algorithms have been developed, including brute force search, dynamic programming, and Monte Carlo algorithms. Unconventional models of computing like DNA computers have also been used to attempt solving the Hamiltonian path problem by exploiting parallel chemical reactions.

1. Hamiltonian path
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2. A game…..
The figure has 20 red
dots. Can you draw a
path that visits each
vertex (Dots) exactly
once????
Try it out….

3. Solution!!!
Icosian game developed by William Rowan Hamilton

4. Hamiltonian path
• In the mathematical field of graph theory, a Hamiltonian path
(or traceable path) is a path in an undirected or directed graph
that visits each vertex exactly once.
A Hamiltonian cycle in a dodecahedron

5. Some definitions….
• A Hamiltonian path or traceable path is a path that visits each
vertex exactly once.
• A graph that contains a Hamiltonian path is called a traceable
graph.
• A graph is Hamiltonian-connected if for every pair of vertices
there is a Hamiltonian path between the two vertices.
• A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph
cycle is a cycle that visits each vertex exactly once (except for
the vertex that is both the start and end, which is visited
twice).
• A graph that contains a Hamiltonian cycle is called a
Hamiltonian graph.

6. Hamiltonian path problem
• Hamiltonian path problem and the Hamiltonian cycle problem
are problems of determining whether a Hamiltonian path (a
path in an undirected or directed graph that visits each vertex
exactly once) or a Hamiltonian cycle exists in a given graph
(whether directed or undirected).

7. Algorithms for solving the problem
• Brute-force search algorithm
There are n! different sequences of vertices that might be
Hamiltonian paths in a given n-vertex graph (and are, in a
complete graph), so a brute force search algorithm that tests all
possible sequences would be very slow. There are several faster
approaches.
• Inclusion–exclusion principle
• Dynamic programming algorithm
• Frank Rubin method
• Monte Carlo algorithm

8. Usingunconventional models of computing
• Because of the difficulty of solving the Hamiltonian path and
cycle problems on conventional computers, they have also
been studied in unconventional models of computing. For
instance, Leonard Adleman showed that the Hamiltonian path
problem may be solved using a DNA computer.
• Exploiting the parallelism inherent in chemical reactions, the
problem may be solved using a number of chemical reaction
steps linear in the number of vertices of the graph; however, it
requires a factorial number of DNA molecules to participate in
the reaction.
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