Metamath Proof Explorer

Definition df-cycls

Description: Define the set of all (simple) cycles (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory) , 3-Oct-2017): "A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex."

According to Bollobas: "If a walk W = x0 x1 ... x(l) is such that l >= 3, x0=x(l), and the vertices x(i), 0 < i < l, are distinct from each other and x0, then W is said to be a cycle." See Definition of Bollobas p. 5.

However, since a walk consisting of distinct vertices (except the first and the last vertex) is a path, a cycle can be defined as path whose first and last vertices coincide. So a cycle is represented by the following sequence: p(0) e(f(1)) p(1) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017) (Revised by AV, 31-Jan-2021)

Ref Expression
Assertion df-cycls ⊢ Cycles = g ∈ V ⟼ f p | f Paths ⁡ g p ∧ p ⁡ 0 = p ⁡ f

Detailed syntax breakdown

Step Hyp Ref Expression
0 ccycls class Cycles
1 vg setvar g
2 cvv class V
3 vf setvar f
4 vp setvar p
5 3 cv setvar f
6 cpths class Paths
7 1 cv setvar g
8 7 6 cfv class Paths ⁡ g
9 4 cv setvar p
10 5 9 8 wbr wff f Paths ⁡ g p
11 cc0 class 0
12 11 9 cfv class p ⁡ 0
13 chash class .
14 5 13 cfv class f
15 14 9 cfv class p ⁡ f
16 12 15 wceq wff p ⁡ 0 = p ⁡ f
17 10 16 wa wff f Paths ⁡ g p ∧ p ⁡ 0 = p ⁡ f
18 17 3 4 copab class f p | f Paths ⁡ g p ∧ p ⁡ 0 = p ⁡ f
19 1 2 18 cmpt class g ∈ V ⟼ f p | f Paths ⁡ g p ∧ p ⁡ 0 = p ⁡ f
20 0 19 wceq wff Cycles = g ∈ V ⟼ f p | f Paths ⁡ g p ∧ p ⁡ 0 = p ⁡ f