Difference between hamiltonian path and euler path

An Euler path is a path that passes through every edge exactly once. If it ends at the initial vertex then it is an Euler cycle.

A Hamiltonian path is a path that passes through every vertex exactly once (NOT every edge). If it ends at the initial vertex then it is a Hamiltonian cycle.

In an Euler path you might pass through a vertex more than once.

In a Hamiltonian path you may not pass through all edges.

Graph Theory Definitions

(In descending order of generality)

• Walk: a sequence of edges where the end of one edge marks the beginning of the next edge

• Trail: a walk which does not repeat any edges. All trails are walks.

• Path: a walk where each vertex is traversed at most once. (paths used to refer to open walks, the definition has changed now) The property of traversing vertices at most once means that edges are also crossed at most once, hence all paths are trails.

Hamiltonian paths & Eulerian trails

• Hamiltonian path: visits every vertex in the graph (exactly once, because it is a path)

• Eulerian trail: visits every edge in the graph exactly once (because it is a trail, vertices may well be crossed more than once.)

Eulerian path must visit each edge exactly once, while Hamiltonian path must visit each vertex exactly once.

They are related but are neither dependent nor mutually exclusive. If a graph has an Eurler cycle, it may or may not also have a Hamiltonian cyle and vice versa.

Euler cycles visit every edge in the graph exactly once. If there are vertices in the graph with more than two edges, then by definition, the cycle will pass through those vertices more than once. As a result, vertices can be repeated but edges cannot.

Hamiltonian cycles visit every vertex in the graph exactly once (similar to the travelling salesman problem). As a result, neither edges nor vertices can be repeated.