What is Eulerian and Hamiltonian graph?
Definition. A cycle that travels exactly once over each edge in a graph is called “Eulerian.” A cycle that travels exactly once over each vertex in a graph is called “Hamiltonian.”
Are all Eulerian graphs Hamiltonian?
An Eulerian graph G (a connected graph in which every vertex has even degree) necessarily has an Euler tour, a closed walk passing through each edge of G exactly once. This tour corresponds to a Hamiltonian cycle in the line graph L(G), so the line graph of every Eulerian graph is Hamiltonian.
What is the difference between Eulerian graph and Eulerian circuit?
An Euler path , in a graph or multigraph, is a walk through the graph which uses every edge exactly once. An Euler circuit is an Euler path which starts and stops at the same vertex.
What is Eulerian graph example?
Euler Graph – A connected graph G is called an Euler graph, if there is a closed trail which includes every edge of the graph G. Euler Path – An Euler path is a path that uses every edge of a graph exactly once. An Euler path starts and ends at different vertices.
What is Hamiltonian graph with example?
Hamiltonian graph – A connected graph G is called Hamiltonian graph if there is a cycle which includes every vertex of G and the cycle is called Hamiltonian cycle. Hamiltonian walk in graph G is a walk that passes through each vertex exactly once.
How do you identify a Hamiltonian graph?
A simple graph with n vertices in which the sum of the degrees of any two non-adjacent vertices is greater than or equal to n has a Hamiltonian cycle.
How do you identify a Eulerian graph?
Eulerian and semi-Eulerian graphs
- If a graph has exactly two vertices of odd degree, then the graph is semi-Eulerian. These two vertices will be the start and the end of the open semi-Eulerian trail.
- If a graph has all even vertices, then the graph is Eulerian. The closed Eulerian trail can start at any vertex.
What is Hamiltonian path example?
A Hamiltonian path, also called a Hamilton path, is a graph path between two vertices of a graph that visits each vertex exactly once. If a Hamiltonian path exists whose endpoints are adjacent, then the resulting graph cycle is called a Hamiltonian cycle (or Hamiltonian cycle).
What is the difference between Euler path and Hamiltonian 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).
How do you know if a graph is Eulerian?
How do you know if a graph is Hamiltonian?