To find the Eulerian path / Eulerian cycle we can use the following strategy: We **find all simple cycles and combine them into one** - this will be the Eulerian cycle. If the graph is such that the Eulerian path is not a cycle, then add the missing edge, find the Eulerian cycle, then remove the extra edge.

If the walk travels along every edge exactly once, then the walk is called an Euler path (or Euler walk ). If, in addition, the starting and ending vertices are the same (so you trace along every edge exactly once and end up where you started), then the walk is called an Euler circuit (or Euler tour ).

Euler's Theorem:

If a graph has more than 2 vertices of odd degree then it has no Euler paths. 2. If a graph is connected and has 0 or exactly 2 vertices of odd degree, then it has at least one Euler path 3. If a graph is connected and has 0 vertices of odd degree, then it has at least one Euler circuit.

Thus, start at one even vertex, travel over each vertex once and only once, and end at the starting point. One example of an Euler circuit for this graph is A, E, A, B, C, B, E, C, D, E, F, D, F, A. This is a circuit that travels over every edge once and only once and starts and ends in the same place.

If a graph has no odd vertices (all even vertices), it has at least one Euler circuit (which, by definition, is also an Euler path). An Euler circuit can start and end at any vertex. If a graph has more than two odd vertices, then it has no Euler paths and no Euler circuits.

Euler's path theorem states the following: 'If a graph has exactly two vertices of odd degree, then it has an Euler path that starts and ends on the odd-degree vertices. Otherwise, it does not have an Euler path. '

Euler or Hamilton Paths

An Euler path is a path that passes through every edge exactly once. If the euler path ends at the same vertex from which is has started it is called as Euler cycle. A Hamiltonian path is a path that passes through every vertex exactly once (NOT every edge).

Euler circuits exist when the degree of all vertices are even. A graph with more than two odd vertices will never have an Euler Path or Circuit. A graph with one odd vertex will have an Euler Path but not an Euler Circuit.

Explanation: Euler's circuit problem can be solved in polynomial time. It can be solved in O(N^{2}).

Fleury's Algorithm

- Start at any vertex if finding an Euler circuit. ...
- Choose any edge leaving your current vertex, provided deleting that edge will not separate the graph into two disconnected sets of edges.
- Add that edge to your circuit, and delete it from the graph.
- Continue until you're done.

Take a graph which is just a cycle on at least 4 vertices, then add an edge between one pair of vertices. Where you added the edge, you will have an odd degree, so the graph cannot have an Eulerian cycle. But the original cycle gives a Hamiltonian cycle.

An Euler path (or Eulerian path ) in a graph G is a simple path that contains every edge of G.

So the total number of Euler circuits is 80.

An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. ▶ An Euler path starts and ends at different vertices. ▶ An Euler circuit starts and ends at the same vertex.

The Euler path is a path, by which we can visit every edge exactly once. We can use the same vertices for multiple times. The Euler Circuit is a special type of Euler path. When the starting vertex of the Euler path is also connected with the ending vertex of that path, then it is called the Euler Circuit.

There are many practical applications to Euler Circuits and Paths. In mathematics, graphs can be used to solve many complex problems, like the Konigsberg Bridge Problem. Moreover, mail carriers can use Eulerian Paths to have a route where they don't have to retrace their previous steps.

Important: An Eulerian circuit traverses every edge in a graph exactly once, but may repeat vertices, while a Hamiltonian circuit visits each vertex in a graph exactly once but may repeat edges.

The Euler tour technique (ETT), named after Leonhard Euler, is a method in graph theory for representing trees. The tree is viewed as a directed graph that contains two directed edges for each edge in the tree.

Answer: The full bipartite graph is non-Hamiltonian but has an Eulerian circuit.

Answer and Explanation: A graph is said to be disconnected if there exist two vertices between there does not exist any path. It has a eulerian path if and only if there are at most two vertices with odd degree.

The Herschel graph is the smallest possible polyhedral graph that does not have a Hamiltonian cycle.

If a graph has an Euler circuit, i.e. a trail which uses every edge exactly once and starts and ends on the same vertex, then it is impossible to also have a trail which uses every edge exactly once and starts and ends on different vertices.

An Eulerian graph G necessarily has an Euler path, a closed walk passing through each edge of G exactly once. This Eulerian path corresponds to a Hamiltonian cycle in the line graph L(G), so the line graph of every Eulerian graph is Hamiltonian.