GRAPH THEORY

1. An Euler path in a graph is a path which traverses each edge of the graph exactly once. 
2. An Euler path which is a cycle is called an Euler cycle. For loop less graphs without isolated vertices, the existence of an Euler path implies the disconnected of the graph, since traversing every edge of such a graph requires visiting each vertex at least once.
3. If a connected graph has an Euler path, one can be constructed by applying Fleury's algorithm. A connected graph has an Euler path if it has exactly zero or two vertices of odd degree. If every vertex has even degree, the graph has an Euler cycle.

CHARACTERISTIC OF EULER CIRCUIT

1. An Euler circuit is a connected graph such that starting at a vertex a , one can traverse along every edge of the graph once to each of the other vertices and return to vertex a .
2. In other words, an Euler circuit is an Euler path that is a circuit. Thus, using the properties of odd and even degree vertices given in the definition of an Euler path, an Euler circuit exists if and only if every vertex of the graph has an even degree.

This graph is an Euler circuit as all vertices have degree 2.

This graph is not an Euler circuit.

CHARACTERISTIC OF HAMILTONIAN PATH

Let G be a graph. A path on G that includes every vertex exactly once is called a Hamiltonian path.

CHARACTERISTIC OF HAMILTONIAN CIRCUIT

Let G be a graph. If there is a cycle visiting all vertices of G exactly once, we say that the cycle is a Hamiltonian cycle. A graph having a Hamiltonian cycle is called a Hamiltonian graph.

EXAMPLE:-

This graph is BOTH Eulerian and Hamiltonian.

This graph is Eulerian, but NOT Hamiltonian.

 
This graph is an Hamiltionian, but NOT Eulerian.

This graph is NEITHER Eulerian NOR Hamiltionian

Comparison of Euler and Hamilton Circuits


There are many differences both in their practical appliances and their theoretical analysis. 

1. First of all the Euler problem takes edges in consideration while the Hamiltonian one has to do with vertices. 
2. The first one tries to pass through every edge of the graph exactly once, while the second one tries to visit all the nodes exactly once without retracing any edge.  

Here is a list with the most important differences between the two problems:-