Metamath Proof Explorer 
< Previous
Next >
Nearby theorems 

Mirrors > Home > MPE Home > Th. List > dfcycl  Structured version Unicode version 
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), 3Oct2017): "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(n1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3Oct2017.) 
Ref  Expression 

dfcycl  Cycles Paths 
Step  Hyp  Ref  Expression 

1  ccycl 24183  . 2 Cycles  
2  vv  . . 3  
3  ve  . . 3  
4  cvv 3113  . . 3  
5  vf  . . . . . . 7  
6  5  cv 1378  . . . . . 6 
7  vp  . . . . . . 7  
8  7  cv 1378  . . . . . 6 
9  2  cv 1378  . . . . . . 7 
10  3  cv 1378  . . . . . . 7 
11  cpath 24176  . . . . . . 7 Paths  
12  9, 10, 11  co 6282  . . . . . 6 Paths 
13  6, 8, 12  wbr 4447  . . . . 5 Paths 
14  cc0 9488  . . . . . . 7  
15  14, 8  cfv 5586  . . . . . 6 
16  chash 12369  . . . . . . . 8  
17  6, 16  cfv 5586  . . . . . . 7 
18  17, 8  cfv 5586  . . . . . 6 
19  15, 18  wceq 1379  . . . . 5 
20  13, 19  wa 369  . . . 4 Paths 
21  20, 5, 7  copab 4504  . . 3 Paths 
22  2, 3, 4, 4, 21  cmpt2 6284  . 2 Paths 
23  1, 22  wceq 1379  1 Cycles Paths 
Colors of variables: wff setvar class 
This definition is referenced by: cycls 24299 cyclispth 24305 cycliscrct 24306 cyclnspth 24307 
Copyright terms: Public domain  W3C validator 