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Quotient graph
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(e6,e7,e8,e4,e7) is not a circuit;
(e1,e6,e7,e8,e4,e5) is a circuit
(e1,e8,e4,e5) is a simple circuit
(e6,e7) is a simple circuit
components of the graph
G1,G2,…,Gω
Proof: e≥n-ω
Let us apply induction on the number of edges of G.
e=0, isolated vertex，has n components ，n=ω,
0=e≥n-ω=0，the result holds
Suppose that result holds for e=e0-1
e=e0, Omitting any edge ,G',
(1)G' has n vertices, ω components and e0-1 edges.
(2)G' has n vertices, ω+1 components and e0-1 edges
2.
If G is connected, then the number of edges of G has at least n-1 edges.
Tree.
Euler paths and circuits, P296 8.2
Hamiltonian paths and circuits, P304 8.3