Cycles in simple graphs

Cycle
A cycle is a path beginning and ending with the same vertex.
Link to original

Length (cycle)
The length of a cycle is the number of edges in it.
Link to original

Simple cycle
A cycle is called simple if it does not contain the same edge twice.
Link to original

Hamiltonian cycle
A Hamiltonian cycle is a [[Simple cycle|simple cycle]] passing through every vertex exactly once.
Link to original

title: Example 1
 
$$
(a,d,c,b,a) \text{ is a simple cycle in } G \text{ of length } 4
$$
 
```graphviz
graph G {	
	a--e
	b--e
	b--f
	c--f
	d--e
	e--f
	
	a--d [penwidth=3]
	c--d [penwidth=3]
	b--c [penwidth=3]
	b--a [penwidth=3]
}

title: Example 1
 
$$
(b,e,d,a,e,b) \text{ is a cycle in } G \text{ of length } 5 \text{ but it is not simple}
$$
 
```graphviz
graph G {	
	a--e [penwidth=3]
	b--e [penwidth=6]
	b--f
	c--f
	d--e [penwidth=3]
	e--f
	a--d [penwidth=3]
	c--d
	b--c
	b--a
}

title: Example 1
 
$$
(c,f,e,d,a,b,c) \text{ is a Hamiltonian cycle in } G
$$
 
```graphviz
graph G {	
	a--e
	b--e
	b--f
	c--f [penwidth=3]
	d--e [penwidth=3]
	e--f [penwidth=3]
	a--d [penwidth=3]
	c--d
	b--c [penwidth=3]
	b--a [penwidth=3]
}