# Hamiltonian Path Problem

October 30, 2019 • ☕️ 1 min read

Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem. One possible Hamiltonian cycle through every vertex of a dodecahedron is shown in red – like all platonic solids, the dodecahedron is Hamiltonian.

## Naive Algorithm

Generate all possible configurations of vertices and print a configuration that satisfies the given constraints. There will be `n!` (n factorial) configurations.

``````while there are untried configurations
{
generate the next configuration
if ( there are edges between two consecutive vertices of this
configuration and there is an edge from the last vertex to
the first ).
{
print this configuration;
break;
}
}``````

## Backtracking Algorithm

Create an empty path array and add vertex `0` to it. Add other vertices, starting from the vertex `1`. Before adding a vertex, check for whether it is adjacent to the previously added vertex and not already added. If we find such a vertex, we add the vertex as part of the solution. If we do not find a vertex then we return false.