An Euler circuit and a Hamiltonian circuit are two types of paths in graph theory that visit all the vertices of a graph. Here are the key differences between them:
1. Euler Circuit:
- An Euler circuit is a circuit that traverses every edge of a graph exactly once and ends at the starting vertex.
- It must visit every edge exactly once.
- It can repeat vertices but not edges.
- Example: In a graph where each vertex has an even degree, an Euler circuit exists.
2. Hamiltonian Circuit:
- A Hamiltonian circuit is a circuit that visits every vertex of a graph exactly once and ends at the starting vertex.
- It must visit every vertex exactly once.
- It cannot repeat vertices but must visit each vertex.
- Example: In a complete graph (a graph where every pair of distinct vertices is connected by a unique edge), a Hamiltonian circuit exists.
In summary, while both circuits visit all vertices, an Euler circuit visits each edge once and may revisit vertices, whereas a Hamiltonian circuit visits each vertex once without revisiting any vertex.
