1. The document discusses Hamiltonian and Euler circuits in graphs. It defines Hamiltonian and Euler paths and circuits, noting that Hamiltonian circuits visit every vertex once while Euler circuits use every edge once. 2. Hamiltonian and Euler problems deal with finding paths between vertices that meet certain criteria. No simple test exists to determine Hamiltonian paths, but certain conditions like having a degree-1 vertex rule out Hamiltonian circuits. 3. Applications of Hamiltonian and Euler problems include computer graphics, circuit design, genome mapping, and transportation planning.

1. Presentation
Hamiltonian And Euler Circuits
Department of Computer Science and Engineering, Amity
University Uttar Pradesh, 2022-2024 Batch
Team Members
1 Gaurav Kumar Gautam(A023144822011)
2 Aditya Prakash Dubey()

2. Graph Theory Basics
Certain graph problems deal with finding a path between two vertices such that
each edge is traversed exactly once, or finding a path between two vertices while
visiting each vertex exactly once. These paths are better known as Euler path and
Hamiltonian path respectively.
The Euler path problem was first proposed in the 1700’s.
Euler paths and circuits :
• An Euler path is a path that uses every edge of a graph exactly once.
• An Euler circuit is a circuit that uses every edge of a graph exactly once.
• An Euler path starts and ends at different vertices.
• An Euler circuit starts and ends at the same vertex.

3. Hamiltonian paths and circuits :
Hamiltonian Path – A simple path in a graph G that passes through every vertex
exactly once is called a Hamiltonian path.
Hamiltonian Circuit – A simple circuit in a graph that passes through every vertex
exactly once is called a Hamiltonian circuit.
Unlike Euler paths and circuits, there is no simple necessary and sufficient criteria to
determine if there are any Hamiltonian paths or circuits in a graph. But there are
certain criteria which rule out the existence of a Hamiltonian circuit in a graph, such
as- if there is a vertex of degree one in a graph then it is impossible for it to have a
Hamiltonian circuit.
There are certain theorems which give sufficient but not necessary conditions for the
existence of Hamiltonian graphs.

4. Applications:
• It is used in various fields such as Computer Graphics, electronic circuit design,
mapping genomes, and operations research.
• A very simple application is planning bus route to pick up students (node->student,
road-> edges, bus path-> Hamiltonian path)
• It is used in genome mapping to combine many tiny fragments of genetic code.S