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Theorem List for Metamath Proof Explorer - 39901-40000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtrlsonfval 39901* The set of trails between two vertices. (Contributed by Alexander van der Vekens, 4-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 15-Jan-2021.) (Revised by AV, 21-Mar-2021.)
Vtx       TrailsOn WalksOn TrailS

Theoremistrlson 39902 Properties of a pair of functions to be a trail between two given vertices. (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 21-Mar-2021.)
Vtx       TrailsOn WalksOn TrailS

Theoremtrlsonprop 39903 Properties of a trail between two vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 16-Jan-2021.)
Vtx       TrailsOn WalksOn TrailS

Theoremtrlsonistrl 39904 A trail between two vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 7-Jan-2021.)
TrailsOn TrailS

Theoremtrlsonwlkon 39905 A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Revised by AV, 7-Jan-2021.)
TrailsOn WalksOn

TheoremtrlOntrl 39906 A trail is a trail between its endpoints. (Contributed by AV, 31-Jan-2021.)
TrailS TrailsOn

21.33.8.17  Paths

Syntaxcpths 39907 Extend class notation with paths (of a graph).
PathS

Syntaxcspths 39908 Extend class notation with simple paths (of a graph).
SPathS

Syntaxcpthson 39909 Extend class notation with paths between two vertices (within a graph).
PathsOn

Syntaxcspthson 39910 Extend class notation with simple paths between two vertices (within a graph).
SPathsOn

Definitiondf-pths 39911* 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 25417).

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.) (Revised by AV, 9-Jan-2021.)

PathS TrailS ..^ ..^

Definitiondf-spths 39912* 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.) (Revised by AV, 9-Jan-2021.)

SPathS TrailS

Definitiondf-pthson 39913* Define the collection of paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 9-Jan-2021.)
PathsOn Vtx Vtx TrailsOn PathS

Definitiondf-spthson 39914* Define the collection of simple paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 9-Jan-2021.)
SPathsOn Vtx Vtx TrailsOn SPathS

Theorempthsfval 39915* The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.)
PathS TrailS ..^ ..^

Theoremspthsfval 39916* The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.)
SPathS TrailS

TheoremisPth 39917 Conditions for a pair of functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.)
PathS TrailS ..^ ..^

TheoremissPth 39918 Conditions for a pair of functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.)
SPathS TrailS

TheoremPthisTrl 39919 A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.)
PathS TrailS

TheoremsPthisPth 39920 A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.)
SPathS PathS

Theorempthis1wlk 39921 A path is a 1-walk (in an undirected graph). (Contributed by AV, 6-Feb-2021.)
PathS 1Walks

Theorempthdivtx 39922 The inner vertices of a path are distinct from all other vertices. (Contributed by AV, 5-Feb-2021.)
PathS ..^

Theorempthdadjvtx 39923 The adjacent vertices of a path of length at least 2 are distinct. (Contributed by AV, 5-Feb-2021.)
PathS ..^

Theorem2pthnloop 39924* A path of length at least 2 does not contain a loop. In contrast, a path of length 1 can contain/be a loop, see lppthon 40039. (Contributed by AV, 6-Feb-2021.)
iEdg       PathS ..^

Theoremupgr2pthnlp 39925* A path of length at least 2 in a pseudograph does not contain a loop. (Contributed by AV, 6-Feb-2021.)
iEdg       UPGraph PathS ..^

Theoremspthdep 39926 A simple path (at least of length 1) has different start and end points (in an undirected graph). (Contributed by AV, 31-Jan-2021.)
SPathS

TheorempthdepissPth 39927 A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 12-Jan-2021.)
PathS SPathS

Theoremupgrwlkdvdelem 39928* Lemma for upgrwlkdvde 39929. Formerly wlkdvspthlem 25416. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Proof shortened by AV, 17-Jan-2021.)
Word ..^

Theoremupgrwlkdvde 39929 In a pseudograph, all edges of a walk consisting of different vertices are different. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspths1wlk 39930. (Contributed by AV, 17-Jan-2021.)
UPGraph 1Walks

Theoremupgrspths1wlk 39930* The set of simple paths in a pseudograph, expressed as walk. Notice that this theorem would not hold for arbitrary hypergraphs, since a walk with distinct vertices does not need to be a trail: let E = { p0, p1, p2 } be a hyperedge, then ( p0, e, p1, e, p2 ) is walk with distinct vertices, but not with distinct edges. Therefore, E is not a trail and, by definition, also no path. (Contributed by AV, 11-Jan-2021.) (Proof shortened by AV, 17-Jan-2021.)
UPGraph SPathS 1Walks

Theoremupgrwlkdvspth 39931 A walk consisting of different vertices is a simple path. Formerly wlkdvspth 25417. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspths1wlk 39930. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Revised by AV, 17-Jan-2021.)
UPGraph 1Walks SPathS

Theorempthsonfval 39932* The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
Vtx       PathsOn TrailsOn PathS

Theoremspthson 39933* The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
Vtx       SPathsOn TrailsOn SPathS

Theoremispthson 39934 Properties of a pair of functions to be a path between two given vertices. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
Vtx       PathsOn TrailsOn PathS

Theoremisspthson 39935 Properties of a pair of functions to be a simple path between two given vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
Vtx       SPathsOn TrailsOn SPathS

Theorempthsonprop 39936 Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 16-Jan-2021.)
Vtx       PathsOn TrailsOn PathS

Theoremspthonprop 39937 Properties of a simple path between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.)
Vtx       SPathsOn TrailsOn SPathS

Theorempthonispth-av 39938 A path between two vertices is a path. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 17-Jan-2021.)
PathsOn PathS

Theorempthontrlon 39939 A path between two vertices is a trail between these vertices. (Contributed by AV, 24-Jan-2021.)
PathsOn TrailsOn

TheorempthOnpth 39940 A path is a path between its endpoints. (Contributed by AV, 31-Jan-2021.)
PathS PathsOn

Theoremisspthonpth-av 39941 A pair of functions is a simple path between two given vertices iff it is a simple path starting and ending at the two vertices. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-Jan-2021.)
Vtx       SPathsOn SPathS

Theoremspthonisspth-av 39942 A simple path between to vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 18-Jan-2021.)
SPathsOn SPathS

Theoremspthonpthon 39943 A simple path between two vertices is a path between these vertices. (Contributed by AV, 24-Jan-2021.)
SPathsOn PathsOn

Theoremspthonepeq-av 39944 The endpoints of a simple path between two vertices are equal iff the path is of length 0. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 18-Jan-2021.)
SPathsOn

Theoremuhgr1wlkspthlem1 39945 Lemma 1 for uhgr1wlkspth 39947. (Contributed by AV, 25-Jan-2021.)
1Walks

Theoremuhgr1wlkspthlem2 39946 Lemma 2 for uhgr1wlkspth 39947. (Contributed by AV, 25-Jan-2021.)
1Walks

Theoremuhgr1wlkspth 39947 Any walk of length 1 between two different vertices is a simple path. (Contributed by AV, 25-Jan-2021.)
Vtx       Edg       WalksOn SPathsOn

Theoremusgr2wlkneq 39948 The vertices and edges are pairwise different in a walk of length 2 in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.)
USGraph 1Walks

Theoremusgr2wlkspthlem1 39949 Lemma 1 for usgr2wlkspth 39951. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.)
1Walks USGraph

Theoremusgr2wlkspthlem2 39950 Lemma 2 for usgr2wlkspth 39951. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.)
1Walks USGraph

Theoremusgr2wlkspth 39951 In a simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.)
USGraph WalksOn SPathsOn

Theorempthdlem1 39952* Lemma 1 for pthd 39955. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 9-Feb-2021.)
Word               ..^ ..^        ..^

Theorempthdlem2lem 39953* Lemma for pthdlem2 39954. (Contributed by AV, 10-Feb-2021.)
Word               ..^ ..^        ..^

Theorempthdlem2 39954* Lemma 2 for pthd 39955. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 10-Feb-2021.)
Word               ..^ ..^        ..^

Theorempthd 39955* Two words representing a trail which also represent a path in a graph. (Contributed by AV, 10-Feb-2021.)
Word               ..^ ..^               TrailS       PathS

21.33.8.18  Closed walks

Syntaxcclwlks 39956 Extend class notation with closed walks (of a graph).
ClWalkS

Definitiondf-clwlks 39957* Define the set of all closed walks (in an undirected graph).

According to definition 4 in [Huneke] p. 2: "A walk of length n on (a graph) G is an ordered sequence v0 , v1 , ... v(n) of vertices such that v(i) and v(i+1) are neighbors (i.e are connected by an edge). We say the walk is closed if v(n) = v0".

According to the definition of a walk as two mappings f from { 0 , ... , ( n - 1 ) } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices, a closed walk is represented by the following sequence: p(0) e(f(0)) p(1) e(f(1)) ... p(n-1) e(f(n-1)) p(n)=p(0).

Notice that by this definition, a single vertex is a closed walk of length 0, see also 0clwlk 25572! (Contributed by Alexander van der Vekens, 12-Mar-2018.) (Revised by AV, 16-Feb-2021.)

ClWalkS 1Walks

TheoremclwlkS 39958* The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.)
ClWalkS 1Walks

TheoremisclWlk 39959 Properties of a pair of functions to be a closed walk (in an undirected graph) in terms of walks. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.)
ClWalkS 1Walks

TheoremisclWlkb 39960 Generalisation of isclwlk 25563: A pair of functions represents a closed walk iff it represents a walk in which the first vertex is equal to the last vertex. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.)
ClWalkS 1Walks

Theoremclwlkis1wlk 39961 A closed walk is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.)
ClWalkS 1Walks

Theoremclwlk1wlk 39962 Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 16-Feb-2021.)
ClWalkS 1Walks

Theoremclwlks1wlks 39963 Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-Feb-2021.)
ClWalkS 1Walks

TheoremisclWlke 39964* Properties of a pair of functions to be a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.)
Vtx       iEdg       ClWalkS Word ..^if-

TheoremclWlkcomp 39965* A closed walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.)
Vtx       iEdg                     ClWalkS Word ..^if-

TheoremclWlkcompim 39966* Implications for the properties of the components of a closed walk. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.)
Vtx       iEdg                     ClWalkS Word ..^if-

21.33.8.19  Circuits and cycles

Syntaxccrcts 39967 Extend class notation with circuits (in a graph).
CircuitS

Syntaxccycls 39968 Extend class notation with cycles (in a graph).
CycleS

Definitiondf-crcts 39969* 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.) (Revised by AV, 31-Jan-2021.)

CircuitS TrailS

Definitiondf-cycls 39970* 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.)

CycleS PathS

TheoremcrctS 39971* The set of circuits (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
CircuitS TrailS

TheoremcyclS 39972* The set of cycles (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
CycleS PathS

TheoremisCrct 39973 Sufficient and necessary conditions for a pair of functions to be a circuit (in an undirected graph): A pair of function "is" (represents) a circuit iff it is a closed trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
CircuitS TrailS

TheoremisCycl 39974 Sufficient and necessary conditions for a pair of functions to be a cycle (in an undirected graph): A pair of function "is" (represents) a cycle iff it is a closed path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
CycleS PathS

Theoremcrctprop 39975 The properties of a circuit: A circuit is a closed trail. (Contributed by AV, 31-Jan-2021.)
CircuitS TrailS

Theoremcyclprop 39976 The properties of a cycle: A cycle is a closed path. (Contributed by AV, 31-Jan-2021.)
CycleS PathS

Theoremcrctisclwlk 39977 A circuit is a closed walk. (Contributed by AV, 17-Feb-2021.)
CircuitS ClWalkS

TheoremcrctisTrl 39978 A circuit is a trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
CircuitS TrailS

TheoremcyclisPth 39979 A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
CycleS PathS

TheoremcyclisWlk 39980 A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 31-Jan-2021.)
CycleS 1Walks

TheoremcyclisCrct 39981 A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
CycleS CircuitS

TheoremcyclnsPth 39982 A (non trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
CycleS SPathS

TheoremcyclisPthon 39983 A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 31-Jan-2021.)
CycleS PathsOn

Theoremlfgrn1cycl 39984* In a loop-free graph there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 2-Feb-2021.)
Vtx       iEdg       CycleS

Theoremumgrn1cycl 39985 In a multigraph graph (with no loops!) there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 2-Feb-2021.)
UMGraph CycleS

Theoremuspgrn2crct 39986 In a simple pseudograph there are no circuits with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 3-Feb-2021.)
USPGraph CircuitS

Theoremusgrn2cycl 39987 In a simple graph there are no cycles with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 4-Feb-2021.)
USGraph CycleS

Theoremcrctcsh1wlkn0lem1 39988 Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 13-Mar-2021.)

Theoremcrctcsh1wlkn0lem2 39989* Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 12-Mar-2021.)
..^

Theoremcrctcsh1wlkn0lem3 39990* Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 12-Mar-2021.)
..^

Theoremcrctcsh1wlkn0lem4 39991* Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 12-Mar-2021.)
..^              cyclShift               Word        ..^if-        ..^ if-

Theoremcrctcsh1wlkn0lem5 39992* Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 12-Mar-2021.)
..^              cyclShift               Word        ..^if-        ..^if-

Theoremcrctcsh1wlkn0lem6 39993* Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 12-Mar-2021.)
..^              cyclShift               Word        ..^if-               if-

Theoremcrctcsh1wlkn0lem7 39994* Lemma for crctcsh1wlkn0 39999. (Contributed by AV, 12-Mar-2021.)
..^              cyclShift               Word        ..^if-               ..^if-

Theoremcrctcshlem1 39995 Lemma for crctcsh 40002. (Contributed by AV, 10-Mar-2021.)
Vtx       iEdg       CircuitS

Theoremcrctcshlem2 39996 Lemma for crctcsh 40002. (Contributed by AV, 10-Mar-2021.)
Vtx       iEdg       CircuitS              ..^       cyclShift

Theoremcrctcshlem3 39997* Lemma for crctcsh 40002. (Contributed by AV, 10-Mar-2021.)
Vtx       iEdg       CircuitS              ..^       cyclShift

Theoremcrctcshlem4 39998* Lemma for crctcsh 40002. (Contributed by AV, 10-Mar-2021.)
Vtx       iEdg       CircuitS              ..^       cyclShift

Theoremcrctcsh1wlkn0 39999* Cyclically shifting the indices of a circuit results in a 1-walk . (Contributed by AV, 10-Mar-2021.)
Vtx       iEdg       CircuitS              ..^       cyclShift               1Walks

Theoremcrctcsh1wlk 40000* Cyclically shifting the indices of a circuit results in a 1-walk . (Contributed by AV, 10-Mar-2021.)
Vtx       iEdg       CircuitS              ..^       cyclShift               1Walks

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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