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Theorem List for Metamath Proof Explorer - 28301-28400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisuvtx 28301* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V UnivVertex  E )  =  { n  e.  V  |  A. k  e.  ( V  \  { n }
 ) { k ,  n }  e.  ran  E } )
 
Theoremuvtxel 28302* An element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( N  e.  ( V UnivVertex  E )  <->  ( N  e.  V  /\  A. k  e.  ( V  \  { N } ) { k ,  N }  e.  ran  E ) ) )
 
Theoremuvtxisvtx 28303 A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( N  e.  ( V UnivVertex  E )  ->  N  e.  V )
 
Theoremuvtx0 28304 There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( (/) UnivVertex  E )  =  (/)
 
Theoremuvtx01vtx 28305* If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. This theorem could have been stated  ( ( V UnivVertex  (/) )  =/=  (/)  <->  ( # `  V
)  =  1 ), but a lot of auxiliary theorems would have been needed. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  (
 ( V UnivVertex  (/) )  =/=  (/)  <->  E. x  V  =  { x } )
 
Theoremuvtxnbgra 28306 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
 |-  (
 ( V USGrph  E  /\  N  e.  ( V UnivVertex  E ) )  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } )
 )
 
Theoremuvtxnm1nbgra 28307 A universal vertex has  n  -  1 neighbors in a graph with  n vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
 |-  (
 ( V USGrph  E  /\  V  e.  Fin )  ->  ( N  e.  ( V UnivVertex  E )  ->  ( # `
  ( <. V ,  E >. Neighbors  N ) )  =  ( ( # `  V )  -  1 ) ) )
 
Theoremuvtxnbgravtx 28308* A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
 |-  (
 ( V USGrph  E  /\  N  e.  ( V UnivVertex  E ) )  ->  A. v  e.  ( V  \  { N } ) N  e.  ( <. V ,  E >. Neighbors  v ) )
 
Theoremcusgrauvtx 28309 In a complete (undirected simple) graph, each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
 |-  ( V ComplUSGrph  E  ->  ( V UnivVertex  E )  =  V )
 
18.23.3.14  Paths and Cycles

Most of the following definitions are based either on Wikipedia articles or on the book "Modern Graph Theory", see Chapter 1 of [Bollobas] p. 1-19.

 
Syntaxcwalk 28310 Extend class notation with Walks (of a graph).
 class Walks
 
Syntaxctrail 28311 Extend class notation with Trails (of a graph).
 class Trails
 
Syntaxcpath 28312 Extend class notation with Paths (of a graph).
 class Paths
 
Syntaxcspath 28313 Extend class notation with Simple Paths (of a graph).
 class SPaths
 
Syntaxcwlkon 28314 PLEASE PUT DESCRIPTION HERE.
 class WalkOn
 
Syntaxctrlon 28315 PLEASE PUT DESCRIPTION HERE.
 class TrailOn
 
Syntaxcpthon 28316 PLEASE PUT DESCRIPTION HERE.
 class PathOn
 
Syntaxccrct 28317 Extend class notation with Circuits (of a graph).
 class Circuits
 
Syntaxccycl 28318 Extend class notation with Cycles (of a graph).
 class Cycles
 
Definitiondf-wlk 28319* Define the set of all Walks (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)."

According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4.

Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

 |- Walks  =  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f
 ) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f
 ) ) ( e `
  ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }
 ) } )
 
Definitiondf-trail 28320* Define the set of all Trails (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A trail is a walk in which all edges are distinct.

According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5.

Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

 |- Trails  =  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v Walks  e
 ) p  /\  Fun  `' f ) } )
 
Definitiondf-pth 28321* Define the set of all Paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

According to Bollobas: "... a path is a walk with distinct vertices.", see Notation of [Bollobas] p. 5. (A walk with distinct vertices is actually a simple path, see wlkdvspth 28366).

Therefore, a path can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, which is injective restricted to the set { 1 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

 |- Paths  =  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v Trails  e
 ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
 ) ) )  /\  ( ( p " { 0 ,  ( # `
  f ) }
 )  i^i  ( p " ( 1..^ ( # `  f ) ) ) )  =  (/) ) }
 )
 
Definitiondf-spth 28322* Define the set of all Simple Paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) (Contributed by Alexander van der Vekens, 20-Oct-2017.)

 |- SPaths  =  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v Trails  e
 ) p  /\  Fun  `' p ) } )
 
Definitiondf-crct 28323* Define the set of all circuits (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A circuit can be a closed walk allowing repetitions of vertices but not edges;"; according to Wikipedia ("Glossary of graph theory terms", https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms, 3-Oct-2017): "A circuit may refer to ... a trail (a closed tour without repeated edges), ...".

Following Bollobas ("A trail whose endvertices coincide (a closed trail) is called a circuit.", see Definition of [Bollobas] p. 5.), a circuit is a closed trail without repeated edges. So the circuit is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) (Contributed by Alexander van der Vekens, 3-Oct-2017.)

 |- Circuits  =  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v Trails  e
 ) p  /\  ( p `  0 )  =  ( p `  ( # `
  f ) ) ) } )
 
Definitiondf-cycl 28324* 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.)

 |- Cycles  =  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v Paths  e
 ) p  /\  ( p `  0 )  =  ( p `  ( # `
  f ) ) ) } )
 
Definitiondf-wlkon 28325* Define the collection of walks with particular endpoints (in an un- directed graph). This corresponds to the "x0-x(l)-walks", see Definition in [Bollobas] p. 5. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
 |- WalkOn  =  ( v  e.  _V ,  e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { <. f ,  p >.  |  ( f ( v Walks  e
 ) p  /\  ( p `  0 )  =  a  /\  ( p `
  ( # `  f
 ) )  =  b ) } ) )
 
Definitiondf-trlon 28326* Define the collection of trails with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
 |- TrailOn  =  ( v  e.  _V ,  e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { <. f ,  p >.  |  ( f ( a ( v WalkOn  e ) b ) p  /\  f
 ( v Trails  e ) p ) } )
 )
 
Definitiondf-pthon 28327* Define the collection of paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
 |- PathOn  =  ( v  e.  _V ,  e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { <. f ,  p >.  |  ( f ( a ( v WalkOn  e ) b ) p  /\  f
 ( v Paths  e ) p ) } )
 )
 
Theoremwlks 28328* The set of walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V Walks  E )  =  { <. f ,  p >.  |  (
 f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
 ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
 ) ) ( E `
  ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }
 ) } )
 
Theoremiswlk 28329* Properties of a pair of functions to be a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( V Walks  E ) P  <->  ( F  e. Word  dom 
 E  /\  P :
 ( 0 ... ( # `
  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
  k ) )  =  { ( P `
  k ) ,  ( P `  (
 k  +  1 ) ) } ) ) )
 
Theorem2mwlk 28330 The two mappings determining a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V ) )
 
Theoremwlkres 28331* Restrictions of walks are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  (
 f ( V W E ) p  ->  f ( V Walks  E ) p )   =>    |-  ( ( V  e.  _V 
 /\  E  e.  _V )  ->  { <. f ,  p >.  |  (
 f ( V W E ) p  /\  ph ) }  e.  _V )
 
Theoremiswlkon 28332 Properties of a pair of functions to be a walk between two given vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 2-Nov-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( F ( A ( V WalkOn  E ) B ) P  <->  ( F ( V Walks  E ) P 
 /\  ( P `  0 )  =  A  /\  ( P `  ( # `
  F ) )  =  B ) ) )
 
Theoremwlkbprop 28333 Basic properties of a walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  ( F ( V Walks  E ) P  ->  ( ( # `  F )  e. 
 NN0  /\  ( V  e.  _V 
 /\  E  e.  _V )  /\  ( F  e.  _V 
 /\  P  e.  _V ) ) )
 
Theoremwlkonwlk 28334 A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.)
 |-  ( F ( V Walks  E ) P  ->  F ( ( P `  0
 ) ( V WalkOn  E ) ( P `  ( # `  F ) ) ) P )
 
Theoremtrls 28335* The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V Trails  E )  =  { <. f ,  p >.  |  ( ( f  e. Word  dom  E  /\  Fun  `' f )  /\  p : ( 0 ... ( # `  f
 ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
 ) ) ( E `
  ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }
 ) } )
 
Theoremistrl 28336* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( V Trails  E ) P 
 <->  ( ( F  e. Word  dom 
 E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 ) ) )
 
Theoremistrl2 28337* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( V Trails  E ) P 
 <->  ( F : ( 0..^ ( # `  F ) ) -1-1-> dom  E  /\  P : ( 0
 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 ) ) )
 
Theoremtrliswlk 28338 A trail is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  ( F ( V Trails  E ) P  ->  F ( V Walks  E ) P )
 
Theoremtrlon 28339* The set of trails between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 4-Nov-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( A ( V TrailOn  E ) B )  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Trails  E ) p ) } )
 
Theoremistrlon 28340 Properties of a pair of functions to be a trail between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 3-Nov-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( F ( A ( V TrailOn  E ) B ) P  <->  ( F ( A ( V WalkOn  E ) B ) P  /\  F ( V Trails  E ) P ) ) )
 
Theoremtrlonprop 28341 A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
 |-  ( F ( A ( V TrailOn  E ) B ) P  ->  ( (
 ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V ) )  /\  ( F ( A ( V WalkOn  E ) B ) P  /\  F ( V Trails  E ) P ) ) )
 
Theoremtrlonwlkon 28342 A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
 |-  ( F ( A ( V TrailOn  E ) B ) P  ->  F ( A ( V WalkOn  E ) B ) P )
 
Theorem0wlk 28343 A pair of an empty set (of edges) and a second set (of vertices) is a walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Walks  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0trl 28344 A pair of an empty set (of edges) and a second set (of vertices) is a trail if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Trails  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theoremwlkntrllem1 28345 Lemma 1 for wlkntrl 28350: F is a word over  {
0 }, the domain of E. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
 |-  V  =  { x ,  y }   &    |-  E  =  { <. 0 ,  { x ,  y } >. }   &    |-  F  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  P  =  { <. 0 ,  x >. , 
 <. 1 ,  y >. , 
 <. 2 ,  x >. }   =>    |-  F  e. Word  dom  E
 
Theoremwlkntrllem2 28346 Lemma 2 for wlkntrl 28350: The cardinality of F is 2. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
 |-  V  =  { x ,  y }   &    |-  E  =  { <. 0 ,  { x ,  y } >. }   &    |-  F  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  P  =  { <. 0 ,  x >. , 
 <. 1 ,  y >. , 
 <. 2 ,  x >. }   =>    |-  ( # `  F )  =  2
 
Theoremwlkntrllem3 28347 Lemma 3 for wlkntrl 28350: P is a function on  (
0 ... 2 ) into  { x ,  y }. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
 |-  V  =  { x ,  y }   &    |-  E  =  { <. 0 ,  { x ,  y } >. }   &    |-  F  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  P  =  { <. 0 ,  x >. , 
 <. 1 ,  y >. , 
 <. 2 ,  x >. }   =>    |-  P : ( 0 ... ( # `  F ) ) --> V
 
Theoremwlkntrllem4 28348* Lemma 4 for wlkntrl 28350: The values of E after F are edges between two vertices enumerated by P. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
 |-  V  =  { x ,  y }   &    |-  E  =  { <. 0 ,  { x ,  y } >. }   &    |-  F  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  P  =  { <. 0 ,  x >. , 
 <. 1 ,  y >. , 
 <. 2 ,  x >. }   =>    |-  A. k  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 
Theoremwlkntrllem5 28349* Lemma 5 for wlkntrl 28350: F is not injective. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
 |-  V  =  { x ,  y }   &    |-  E  =  { <. 0 ,  { x ,  y } >. }   &    |-  F  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  P  =  { <. 0 ,  x >. , 
 <. 1 ,  y >. , 
 <. 2 ,  x >. }   =>    |-  -. 
 Fun  `' F
 
Theoremwlkntrl 28350* A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one edge to the other is a walk, but not a trail. Notice that  <. V ,  E >. is a simple graph (without loops) only if  x  =/=  y. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
 |-  V  =  { x ,  y }   &    |-  E  =  { <. 0 ,  { x ,  y } >. }   &    |-  F  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  P  =  { <. 0 ,  x >. , 
 <. 1 ,  y >. , 
 <. 2 ,  x >. }   =>    |-  ( F ( V Walks  E ) P  /\  -.  F ( V Trails  E ) P )
 
Theoremusgrnloop 28351* In an undirected simple graph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.)
 |-  (
 ( V USGrph  E  /\  F ( V Walks  E ) P )  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1
 ) ) )
 
Theorempths 28352* The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V Paths  E )  =  { <. f ,  p >.  |  (
 f ( V Trails  E ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
 ) ) )  /\  ( ( p " { 0 ,  ( # `
  f ) }
 )  i^i  ( p " ( 1..^ ( # `  f ) ) ) )  =  (/) ) }
 )
 
Theoremspths 28353* The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V SPaths  E )  =  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  Fun  `' p ) } )
 
Theoremispth 28354 Properties of a pair of functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( V Paths  E ) P  <->  ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F ) ) )  /\  ( ( P " { 0 ,  ( # `
  F ) }
 )  i^i  ( P " ( 1..^ ( # `  F ) ) ) )  =  (/) ) ) )
 
Theoremisspth 28355 Properties of a pair of functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( V SPaths  E ) P 
 <->  ( F ( V Trails  E ) P  /\  Fun  `' P ) ) )
 
Theorem0pth 28356 A pair of an empty set (of edges) and a second set (of vertices) is a path if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Paths  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0spth 28357 A pair of an empty set (of edges) and a second set (of vertices) is a simple path if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V SPaths  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorempthistrl 28358 A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  ( F ( V Paths  E ) P  ->  F ( V Trails  E ) P )
 
Theoremspthispth 28359 A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  ( F ( V SPaths  E ) P  ->  F ( V Paths  E ) P )
 
Theorempthdepisspth 28360 A path with different start end end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  (
 ( F ( V Paths  E ) P  /\  ( P `  0 )  =/=  ( P `  ( # `  F ) ) )  ->  F ( V SPaths  E ) P )
 
Theorempthon 28361* The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( A ( V PathOn  E ) B )  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Paths  E ) p ) } )
 
Theoremispthon 28362 Properties of a pair of functions to be a path between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( F ( A ( V PathOn  E ) B ) P  <->  ( F ( A ( V WalkOn  E ) B ) P  /\  F ( V Paths  E ) P ) ) )
 
Theoremredwlklem 28363 Lemma for redwlk 28364. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  (
 ( F ( V Walks  E ) P  /\  1  <_  ( # `  F ) )  ->  ( # `  ( F  |`  ( 0..^ ( ( # `  F )  -  1 ) ) ) )  =  ( ( # `  F )  -  1 ) )
 
Theoremredwlk 28364 A walk ending at the last but one vertex of the walk is a walk. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  (
 ( F ( V Walks  E ) P  /\  1  <_  ( # `  F ) )  ->  ( F  |`  ( 0..^ ( ( # `  F )  -  1 ) ) ) ( V Walks  E ) ( P  |`  ( 0..^ ( # `  F ) ) ) )
 
Theoremwlkdvspthlem 28365* Lemma for wlkdvspth 28366. (Contributed by Alexander van der Vekens, 27-Oct-2017.)
 |-  (
 ( F  e. Word  dom  E 
 /\  P : ( 0 ... ( # `  F ) ) -1-1-> V  /\  A. k  e.  (
 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 )  ->  Fun  `' F )
 
Theoremwlkdvspth 28366 A walk consisting of different vertices is a simple path. (Contributed by Alexander van der Vekens, 27-Oct-2017.)
 |-  (
 ( F ( V Walks  E ) P  /\  Fun  `' P )  ->  F ( V SPaths  E ) P )
 
Theoremcrcts 28367* The set of circuits (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V Circuits  E )  =  { <. f ,  p >.  |  ( f ( V Trails  E ) p  /\  ( p `  0 )  =  ( p `  ( # `  f ) ) ) } )
 
Theoremcycls 28368* The set of cycles (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V Cycles  E )  =  { <. f ,  p >.  |  ( f ( V Paths  E ) p 
 /\  ( p `  0 )  =  ( p `  ( # `  f
 ) ) ) }
 )
 
Theoremiscrct 28369 Properties of a pair of functions to be a circuit (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( V Circuits  E ) P 
 <->  ( F ( V Trails  E ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
 
Theoremiscycl 28370 Properties of a pair of functions to be a cycle (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z ) )  ->  ( F ( V Cycles  E ) P 
 <->  ( F ( V Paths  E ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
 
Theorem0crct 28371 A pair of an empty set (of edges) and a second set (of vertices) is a circuit if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Circuits  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0cycl 28372 A pair of an empty set (of edges) and a second set (of vertices) is a cycle if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Cycles  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theoremcrctistrl 28373 A circuit is a trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( F ( V Circuits  E ) P  ->  F ( V Trails  E ) P )
 
Theoremcyclispth 28374 A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( F ( V Cycles  E ) P  ->  F ( V Paths  E ) P )
 
Theoremcycliscrct 28375 A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( F ( V Cycles  E ) P  ->  F ( V Circuits  E ) P )
 
Theoremcyclnspth 28376 A (non trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( F  =/=  (/)  ->  ( F ( V Cycles  E ) P 
 ->  -.  F ( V SPaths  E ) P ) )
 
Theoremcycliswlk 28377 A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.)
 |-  ( F ( V Cycles  E ) P  ->  F ( V Walks  E ) P )
 
Theoremcyclispthon 28378 A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017.)
 |-  ( F ( V Cycles  E ) P  ->  F ( ( P `  0
 ) ( V PathOn  E ) ( P `  0 ) ) P )
 
Theoremfargshiftlem 28379 If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
 |-  (
 ( N  e.  NN0  /\  X  e.  ( 0..^ N ) )  ->  ( X  +  1
 )  e.  ( 1
 ... N ) )
 
Theoremfargshiftfv 28380* If a class is a function, then the values of the "shifted function" correspond to the function values of the class. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
 |-  G  =  ( x  e.  (
 0..^ ( # `  F ) )  |->  ( F `
  ( x  +  1 ) ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) --> dom  E )  ->  ( X  e.  ( 0..^ N )  ->  ( G `  X )  =  ( F `  ( X  +  1
 ) ) ) )
 
Theoremfargshiftf 28381* If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
 |-  G  =  ( x  e.  (
 0..^ ( # `  F ) )  |->  ( F `
  ( x  +  1 ) ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) --> dom  E )  ->  G : ( 0..^ ( # `  F ) ) --> dom  E )
 
Theoremfargshiftf1 28382* If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
 |-  G  =  ( x  e.  (
 0..^ ( # `  F ) )  |->  ( F `
  ( x  +  1 ) ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) -1-1-> dom  E )  ->  G :
 ( 0..^ ( # `  F ) ) -1-1-> dom  E )
 
Theoremfargshiftfo 28383* If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
 |-  G  =  ( x  e.  (
 0..^ ( # `  F ) )  |->  ( F `
  ( x  +  1 ) ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) -onto-> dom 
 E )  ->  G : ( 0..^ ( # `  F ) )
 -onto->
 dom  E )
 
Theoremfargshiftfva 28384* The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
 |-  G  =  ( x  e.  (
 0..^ ( # `  F ) )  |->  ( F `
  ( x  +  1 ) ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) --> dom  E )  ->  ( A. k  e.  ( 1 ... N ) ( E `  ( F `  k ) )  =  [_ k  /  x ]_ P  ->  A. l  e.  ( 0..^ N ) ( E `
  ( G `  l ) )  = 
 [_ ( l  +  1 )  /  x ]_ P ) )
 
Theoremeupatrl 28385* An Eulerian path is a trail.

Unfortunately, the edge function  F of an Eulerian path has the domain  ( 1 ... ( # `  F
) ), whereas the edge functions of all kinds of walks defined here have the domain  ( 0..^ ( # `  F
) ) (i.e. the edge functions are "words of edge indices", see discussion and proposal of Mario Carneiro at https://groups.google.com/d/msg/metamath/KdVXdL3IH3k/2-BYcS_ACQAJ). Therefore, the arguments of the edge function of an Eulerian path must be shifted by 1 to obtain an edge function of a trail in this theorem, using the auxiliary theorems above (fargshiftlem 28379, fargshiftfv 28380, etc.). The definition of an Eulerian path and all related theorems should be modified as soon as the graph theory is integrated into the main part of set.mm. (Contributed by Alexander van der Vekens, 24-Nov-2017.)

 |-  G  =  ( x  e.  (
 0..^ ( # `  F ) )  |->  ( F `
  ( x  +  1 ) ) )   =>    |-  ( F ( V EulPaths  E ) P  ->  G ( V Trails  E ) P )
 
Theoremusgrcyclnl1 28386 In an undirected simple graph (with no loops!) there are no cycles with length 1 (consisting of one edge ). (Contributed by Alexander van der Vekens, 7-Nov-2017.)
 |-  (
 ( V USGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `
  F )  =/=  1 )
 
Theoremusgrcyclnl2 28387 In an undirected simple graph (with no loops!) there are no cycles with length 2 (consisting of two edges ). (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  (
 ( V USGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `
  F )  =/=  2 )
 
Theorem3cycl3dv 28388 In a simple graph, the vertices of a 3-cycle are mutually different. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  (
 ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A ) )
 
Theoremnvnencycllem 28389 Lemma for 3v3e3cycl1 28390 and 4cycl4v4e 28412. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  (
 ( ( Fun  E  /\  F  e. Word  dom  E ) 
 /\  ( X  e.  NN0  /\  X  <  ( # `  F ) ) ) 
 ->  ( ( E `  ( F `  X ) )  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )
 
Theorem3v3e3cycl1 28390* If there is a cycle of length 3 in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  (
 ( Fun  E  /\  F ( V Cycles  E ) P  /\  ( # `  F )  =  3 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E 
 /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
 
Theoremconstr3lem1 28391 Lemma for constr3trl 28405 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( F  e.  _V 
 /\  P  e.  _V )
 
Theoremconstr3lem2 28392 Lemma for constr3trl 28405 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( # `  F )  =  3
 
Theoremconstr3lem4 28393 Lemma for constr3trl 28405 etc. The proof could be much shorter if a theorem "fvprg" analogous to fvsng 5730, fvpr1 5738 and fvpr2 5739 was available. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( ( P `  0
 )  =  A  /\  ( P `  1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3
 )  =  A ) ) )
 
Theoremconstr3lem5 28394 Lemma for constr3trl 28405 etc. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( F `
  0 )  =  ( `' E `  { A ,  B }
 )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E ` 
 { C ,  A } ) )
 
Theoremconstr3lem6 28395 Lemma for constr3pthlem3 28403. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A ) ) 
 ->  ( { ( P `
  0 ) ,  ( P `  3
 ) }  i^i  {
 ( P `  1
 ) ,  ( P `
  2 ) }
 )  =  (/) )
 
Theoremconstr3trllem1 28396 Lemma for constr3trl 28405. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  F  e. Word  dom  E )
 
Theoremconstr3trllem2 28397 Lemma for constr3trl 28405. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  Fun  `' F )
 
Theoremconstr3trllem3 28398 Lemma for constr3trl 28405. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P :
 ( 0 ... ( # `
  F ) ) --> V )
 
Theoremconstr3trllem4 28399 Lemma for constr3trl 28405. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P :
 ( 0 ... 3
 ) --> V )
 
Theoremconstr3trllem5 28400* Lemma for constr3trl 28405. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  A. k  e.  (
 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 )
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