Hamiltonian Cycle

Hamiltonian Cycle

Definition and Basic Concepts

• A Hamiltonian Cycle is a closed loop on a graph where every node (vertex) is visited exactly once.
• The cycle starts and ends at the same vertex.
• The name comes from the Irish mathematician Sir William Rowan Hamilton.

Properties of Hamiltonian Cycle

• A graph with a Hamiltonian cycle is called a Hamiltonian graph.
• A Hamiltonian cycle is a special case of a Hamiltonian path. The difference is that a path does not have to return to the original vertex, while a cycle does.
• All complete graphs (graphs where every pair of distinct vertices is connected by a unique edge) with five or more vertices are Hamiltonian.

Hamiltonian Cycle Problem

• The Hamiltonian Cycle Problem is a critical problem in graph theory and computer science. It is focused on whether a given graph contains a Hamiltonian Cicrcle or not.
• This problem is a NP-complete problem; it may require a significant amount of computational resources to solve for large graphs.

Algorithms to Find Hamiltonian Cycles

• Various algorithms, like the backtracking algorithm, can find Hamiltonian cycles in a graph. However, these solutions may have exponential time complexity.
• The Hamiltonian Cycle algorithm involves checking all the vertices and avoiding those already included in the path until we reach the starting vertex.
• The Nearest Neighbour Algorithm is a heuristic algorithm that can find an approximate Hamiltonian cycle for weighted graphs.

Applications of Hamiltonian Cycle

• Real-world applications of the Hamiltonian cycle include the Travelling Salesman Problem, which aims to find the shortest possible route that visits each city exactly once and returns to the origin city.
• Hamiltonian cycles also apply to data routing in computer networks to avoid data packet loss and make efficient use of network resources.