Can a graph have a Euler circuit and a Hamiltonian circuit?
A circuit is any path in the graph which begins and ends at the same vertex. The whole subject of graph theory started with Euler and the famous Konisberg Bridge Problem. An Eulerian circuit passes along each edge once and only once, and a Hamiltonian circuit visits each vertex once and only once.
Does there exist a graph which has Hamiltonian circuit without Hamiltonian path?
But there are certain criteria which rule out the existence of a Hamiltonian circuit in a graph, such as- if there is a vertex of degree one in a graph then it is impossible for it to have a Hamiltonian circuit.
Do all complete graphs have Hamiltonian cycles?
Every complete graph with more than two vertices is a Hamiltonian graph. This follows from the definition of a complete graph: an undirected, simple graph such that every pair of nodes is connected by a unique edge. The graph of every platonic solid is a Hamiltonian graph.
Does the graph have a Euler circuit?
How could we have an Euler circuit? Thus for a graph to have an Euler circuit, all vertices must have even degree. The converse is also true: if all the vertices of a graph have even degree, then the graph has an Euler circuit, and if there are exactly two vertices with odd degree, the graph has an Euler path.
How do you know if a graph has a Hamiltonian circuit?
- A connected graph is said to have a Hamiltonian circuit if it has a circuit that ‘visits’ each node (or vertex) exactly once.
- For instance, the graph below has 20 nodes.
- The red lines show a Hamiltonian circuit that this graph contains.
- So by definition, this is a Hamiltonian graph.
What is the difference between Hamiltonian graph and Euler graphs?
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.
Are all Euler 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.
How do you tell if a graph has a Hamiltonian circuit?
Which graph will have a Hamiltonian circuit?
A Hamiltonian path or traceable path is a path that visits each vertex of the graph exactly once. A graph that contains a Hamiltonian path is called a traceable graph. A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices.