An Euler graph may be defined as-

__Euler Graph Example-__

__Euler Graph Example-__

The following graph is an example of an Euler graph-

Here,

· This graph is a connected graph and all its vertices are of even degree.

· Therefore, it is an Euler graph.

Alternatively, the above graph contains an Euler circuit BACEDCB, so it is an Euler graph.

**Also Read-** __Planar Graph__

__Euler Path-__

__Euler Path-__

Euler path is also known as **Euler Trail** or **Euler Walk**.

· If there exists a ** Trail** in the connected graph that contains all the edges of the graph, then that trail is called as an Euler trail.

**OR**

· If there exists a walk in the connected graph that visits every edge of the graph exactly once with or without repeating the vertices, then such a walk is called as an Euler walk.

A graph will contain an Euler path if and only if it contains at most two vertices of odd degree.NOTE |

__Euler Path Examples-__

__Euler Path Examples-__

Examples of Euler path are as follows-

__Euler Circuit-__

__Euler Circuit-__

Euler circuit is also known as **Euler Cycle** or **Euler Tour**.

· If there exists a ** Circuit** in the connected graph that contains all the edges of the graph, then that circuit is called as an Euler circuit.

**OR**

· If there exists a walk in the connected graph that starts and ends at the same vertex and visits every edge of the graph exactly once with or without repeating the vertices, then such a walk is called as an Euler circuit.

**OR**

· An Euler trail that starts and ends at the same vertex is called as an Euler circuit.

**OR**

· A closed Euler trail is called as an Euler circuit.

A graph will contain an Euler circuit if and only if all its vertices are of even degree.NOTE |

__Euler Circuit Examples-__

__Euler Circuit Examples-__

Examples of Euler circuit are as follows-

__Semi-Euler Graph-__

__Semi-Euler Graph-__

If a connected graph contains an Euler trail but does not contain an Euler circuit, then such a graph is called as a semi-Euler graph.

Thus, for a graph to be a semi-Euler graph, following two conditions must be satisfied-

· Graph must be connected.

· Graph must contain an Euler trail.

__Example-__

__Example-__

Here,

· This graph contains an Euler trail BCDBAD.

· But it does not contain an Euler circuit.

· Therefore, it is a semi-Euler graph.

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