Euler's path and Hamiltonian cycle are both types of graph theory problems. An Euler's path is a path in a graph that visits each vertex exactly once and ends at the starting vertex. A Hamiltonian cycle is a cycle in a graph that visits each vertex exactly once and ends at the starting vertex.
No, they are not the same thing. While both involve visiting each vertex exactly once, an Euler's path is a path while a Hamiltonian cycle is a cycle.
Yes, it is possible for an Euler's path to be the same as a Hamiltonian cycle. This occurs when the starting vertex is also the ending vertex, resulting in a cycle that includes all vertices in the graph.
There are several algorithms and criteria that can be used to determine if a graph has an Euler's path or Hamiltonian cycle. For an Euler's path, the graph must have exactly 0 or 2 vertices with an odd degree. For a Hamiltonian cycle, there is no known efficient algorithm, but it can be determined by trying all possible paths or cycles.
Euler's path and Hamiltonian cycle have many real-world applications, particularly in transportation and logistics. For example, they can be used to optimize delivery routes or plan efficient travel itineraries. They are also used in computer networks to find the shortest path between nodes. Additionally, Euler's path and Hamiltonian cycle have applications in chemistry, biology, and other fields for understanding molecular structures and interactions.