Give two examples of graphs that have Euler circuits and Hamiltonian circuits that are not the same.

Give two examples of graphs that have Euler circuits and Hamiltonian circuits that are not the same.

Given information:

Graphs that have Euler circuits and Hamiltonian circuits that are not the same.

Calculation:

We know that:

The degree of a vertex is the number of edges that connect to the vertex.

Note: a loop at a vertex counts as two edges.

An Euler circuit is a circuit that contains edge of the graph.

A Hamiltonian circuit is a simple circuit that passes through every vertex exactly once.

A connected graph has an Euler circuit if and only if each of the vertices has an even degree.

Two examples of graphs with Euler circuits and Hamilton circuits are given in the image below. We also note that the Euler circuits and Hamilton circuits are not the same.

The two graphs below do contain an Euler circuit, because they are both connected graphs that contain vertices with even degrees only. For example, e1e2e5e6e3e4 is an Euler circuit in the first graph and e7e8