[algorithm] Difference between hamiltonian path and euler path

Can some one tell me the difference between hamiltonian path and euler path. They seem similar!

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.)

I'll use a common example in biology; reconstructing a genome by making DNA samples.

De-novo assembly

To construct a genome from short reads, it's necessary to construct a graph of those reads. We do it by breaking the reads into k-mers and assemble them into a graph.

We can reconstruct the genome by visiting each node once as in the diagram. This is known as Hamiltonian path.

Unfortunately, constructing such path is NP-hard. It's not possible to derive an efficient algorithm for solving it. Instead, in bioinformatics we construct a Eulerian cycle where an edge represents an overlap.

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.

Euler path is a graph using every edge(NOTE) of the graph exactly once. Euler circuit is a euler path that returns to it starting point after covering all edges.

While hamilton path is a graph that covers all vertex(NOTE) exactly once. When this path returns to its starting point than this path is called hamilton circuit.

Euler Path - An Euler path is a path in which each edge is traversed exactly once.

Hamiltonian Path - An Hamiltonian path is path in which each vertex is traversed exactly once.

If you have ever confusion remember E - Euler E - Edge.

An Euler path is a path that uses every edge of a graph exactly once.and it must have exactly two odd vertices.the path starts and ends at different vertex. A Hamiltonian cycle is a cycle that contains every vertex of the graph hence you may not use all the edges of the graph.

A Hamiltonian path visits every node (or vertex) exactly once, and a Eulerian path traverses every edge exactly once.