## Can a graph have a Euler circuit but not a Hamiltonian circuit?

A Hamiltonian cycle is a cycle that visits every vertex of the graph exactly once. An Eulerian graph is one which has an Eulerian cycle. An Eulerian cycle is a trail that starts and ends on the same vertex visiting every edge in the graph exactly once. Nope.

## Are all Eulerian circuits Hamiltonian?

Originally Answered: Is every Eulerian graph Hamilton? No. An Eulerian graph must have a trail that uses every EDGE in the graph and starts and ends on the same vertex. A Hamiltonian graph must contain a walk that visits every VERTEX (except for the initial/ending vertex) exactly once.

**What is Eulerian not Hamiltonian?**

The complete bipartite graph K2,4 has an Eulerian circuit, but is non-Hamiltonian (in fact, it doesn’t even contain a Hamiltonian path). Any Hamiltonian path would alternate colors (and there’s not enough blue vertices). Since every vertex has even degree, the graph has an Eulerian circuit.

### How do you prove there is no Hamiltonian circuit?

Proving a graph has no Hamiltonian cycle [closed]

- A graph with a vertex of degree one cannot have a Hamilton circuit.
- Moreover, if a vertex in the graph has degree two, then both edges that are incident with this vertex must be part of any Hamilton circuit.
- A Hamilton circuit cannot contain a smaller circuit within it.

### Which graph will have a Hamiltonian circuit?

By the way if a graph has a Hamilton circuit then it has a Hamilton path. Just do not go back to home. Complete Graph: A complete graph is a graph with N vertices in which every pair of vertices is joined by exactly one edge….6.4: Hamiltonian Circuits.

Hamilton Circuit | Mirror Image | Total Weight (Miles) |
---|---|---|

ABDCA | ACDBA | 20 |

ACBDA | ADBCA | 20 |

**Can a graph be Eulerian and Hamiltonian?**

A connected graph G is Eulerian if there is a closed trail which includes every edge of G, such a trail is called an Eulerian trail. A connected graph G is Hamiltonian if there is a cycle which includes every vertex of G; such a cycle is called a Hamiltonian cycle. This graph is BOTH Eulerian and Hamiltonian.

#### How do you know if its a Hamiltonian circuit?

A Hamiltonian circuit is a circuit that visits every vertex once with no repeats. Being a circuit, it must start and end at the same vertex. A Hamiltonian path also visits every vertex once with no repeats, but does not have to start and end at the same vertex.

#### What is the difference between Euler circuit and Hamiltonian circuit?

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.

**Is Herschel graph Hamiltonian?**

As a bipartite graph that has an odd number of vertices, the Herschel graph does not contain a Hamiltonian cycle (a cycle of edges that passes through each vertex exactly once). Thus, a cycle passing once through each of the eleven vertices cannot exist in the Herschel graph.

## How to find Euler circuit?

Start from the source node,call it as current node u.

## What makes an Euler circuit?

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. In other words, an Euler circuit is an Euler path that is a circuit.

**What is an Euler path?**

An Euler path in a graph is a path which traverses each edge of the graph exactly once. An Euler path which is a cycle is called an Euler cycle.