For the following sequences of vertices, determine whether each sequence of vertices is a circuit, whether it is an Euler circuit, and whether it is a

Hamilton circuit. Justify your answers.

Complete parts $($a$)$ through $($c$)$ below.

$($a$)$ Consider the sequence of vertices $BCDEAB.$ Select the correct choice and, if necessary, fill in the answer box$($es$)$ to complete your choice.

A$.$ The sequence is a circuit and an Euler circuit, but not a Hamilton circuit because the circuit visits nonstarting vertex

more than once.

B$.$ The sequence is a circuit, but it is neither an Euler circuit nor a Hamilton circuit because the circuit does not use edge

and it visits nonstarting vertex

more

than once.

C$.$ The sequence is a circuit, an Euler circuit, and a Hamilton circuit because each edge is used exactly once, and the sequence visits each vertex exactly once and

returns to the starting vertex.

D$.$ The sequence is not a circuit because it includes at least one edge that does not exist on the graph, or uses an edge more than once. Since it is not a circuit, it

cannot be an Euler circuit or a Hamilton circuit.

E$.$ The sequence is a circuit and a Hamilton circuit, but not an Euler circuit because the circuit does not use edge