Paths in simple graphs

Path
A path $p = ⟨ v_{1}, v_{2}, \dots, v_{k}, v_{k + 1} ⟩$ is a sequence of vertices where $(v_{i}, v_{i + 1})$ is an edge for each $i = 1, 2, \dots, k$ .
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Length (path)
The length of a path is the number of edges in it.
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Simple path
A path is called simple if it does not contain the same edge twice.
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Hamiltonian path
A Hamiltonian path is a [[Simple path|simple path]] passing through every vertex exactly once.
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title: Example 1
 
$$
(a,b,c,f,b,e) \text{ is a simple path in } G \text{ of length } 5
$$
 
```graphviz
graph G {
	a--d
	a--e
	d--c
	d--e
	e--f
	
	edge [colorscheme=orrd9]
	a--b [color=9 penwidth=3]
	b--c [color=8 penwidth=3]
	c--f [color=7 penwidth=3]
	f--b [color=6 penwidth=3]
	b--e [color=5 penwidth=3]
}

title: Example 2
 
$$
(a,b,c,d,a,b) \text{ is a path in } G \text{ of length } 5 \text{ but it is not simple as it contains } a-b \text{ twice}
$$
 
```graphviz
graph G {
	a--e
	d--c
	e--f
	b--c
	c--f
	f--b
	
	edge [colorscheme=greens9]
	a--b [color=9 penwidth=5]
	b--e [color=8 penwidth=3]
	d--e [color=7 penwidth=3]
	a--d [color=6 penwidth=3]
}

title: Example 3
 
$$
(d,a,e,b,f,c) \text{ is a Hamiltonian path in } G
$$
 
```graphviz
graph G {
	d--c
	e--f
	b--c
	a--b
	d--e
	
	edge [colorscheme=blues9]
	a--d [color=9 penwidth=3]
	a--e [color=8 penwidth=3]
	b--e [color=7 penwidth=3]
	f--b [color=6 penwidth=3]
	c--f [color=5 penwidth=3]
}