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Theorem List for Metamath Proof Explorer - 29001-29100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempthontrlon 29001 A path between two vertices is a trail between these vertices. (Contributed by AV, 24-Jan-2021.)
(𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(TrailsOnβ€˜πΊ)𝐡)𝑃)
 
Theorempthonpth 29002 A path is a path between its endpoints. (Contributed by AV, 31-Jan-2021.)
(𝐹(Pathsβ€˜πΊ)𝑃 β†’ 𝐹((π‘ƒβ€˜0)(PathsOnβ€˜πΊ)(π‘ƒβ€˜(β™―β€˜πΉ)))𝑃)
 
Theoremisspthonpth 29003 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.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉) ∧ (𝐹 ∈ π‘Š ∧ 𝑃 ∈ 𝑍)) β†’ (𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃 ↔ (𝐹(SPathsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
 
Theoremspthonisspth 29004 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.)
(𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(SPathsβ€˜πΊ)𝑃)
 
Theoremspthonpthon 29005 A simple path between two vertices is a path between these vertices. (Contributed by AV, 24-Jan-2021.)
(𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃 β†’ 𝐹(𝐴(PathsOnβ€˜πΊ)𝐡)𝑃)
 
Theoremspthonepeq 29006 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.) (Proof shortened by AV, 31-Oct-2021.)
(𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃 β†’ (𝐴 = 𝐡 ↔ (β™―β€˜πΉ) = 0))
 
Theoremuhgrwkspthlem1 29007 Lemma 1 for uhgrwkspth 29009. (Contributed by AV, 25-Jan-2021.)
((𝐹(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 1) β†’ Fun ◑𝐹)
 
Theoremuhgrwkspthlem2 29008 Lemma 2 for uhgrwkspth 29009. (Contributed by AV, 25-Jan-2021.)
((𝐹(Walksβ€˜πΊ)𝑃 ∧ ((β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) ∧ ((π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) β†’ Fun ◑𝑃)
 
Theoremuhgrwkspth 29009 Any walk of length 1 between two different vertices is a simple path. (Contributed by AV, 25-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.) (Revised by AV, 7-Jul-2022.)
((𝐺 ∈ π‘Š ∧ (β™―β€˜πΉ) = 1 ∧ 𝐴 β‰  𝐡) β†’ (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ↔ 𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃))
 
Theoremusgr2wlkneq 29010 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.)
(((𝐺 ∈ USGraph ∧ 𝐹(Walksβ€˜πΊ)𝑃) ∧ ((β™―β€˜πΉ) = 2 ∧ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ (((π‘ƒβ€˜0) β‰  (π‘ƒβ€˜1) ∧ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2) ∧ (π‘ƒβ€˜1) β‰  (π‘ƒβ€˜2)) ∧ (πΉβ€˜0) β‰  (πΉβ€˜1)))
 
Theoremusgr2wlkspthlem1 29011 Lemma 1 for usgr2wlkspth 29013. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.)
((𝐹(Walksβ€˜πΊ)𝑃 ∧ (𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ Fun ◑𝐹)
 
Theoremusgr2wlkspthlem2 29012 Lemma 2 for usgr2wlkspth 29013. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.)
((𝐹(Walksβ€˜πΊ)𝑃 ∧ (𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜(β™―β€˜πΉ)))) β†’ Fun ◑𝑃)
 
Theoremusgr2wlkspth 29013 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.) (Proof shortened by AV, 31-Oct-2021.)
((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2 ∧ 𝐴 β‰  𝐡) β†’ (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ↔ 𝐹(𝐴(SPathsOnβ€˜πΊ)𝐡)𝑃))
 
Theoremusgr2trlncl 29014 In a simple graph, any trail of length 2 does not start and end at the same vertex. (Contributed by AV, 5-Jun-2021.) (Proof shortened by AV, 31-Oct-2021.)
((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝐹(Trailsβ€˜πΊ)𝑃 β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2)))
 
Theoremusgr2trlspth 29015 In a simple graph, any trail of length 2 is a simple path. (Contributed by AV, 5-Jun-2021.)
((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝐹(Trailsβ€˜πΊ)𝑃 ↔ 𝐹(SPathsβ€˜πΊ)𝑃))
 
Theoremusgr2pthspth 29016 In a simple graph, any path of length 2 is a simple path. (Contributed by Alexander van der Vekens, 25-Jan-2018.) (Revised by AV, 5-Jun-2021.)
((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 ↔ 𝐹(SPathsβ€˜πΊ)𝑃))
 
Theoremusgr2pthlem 29017* Lemma for usgr2pth 29018. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))
 
Theoremusgr2pth 29018* In a simple graph, there is a path of length 2 iff there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.) (Proof shortened by AV, 31-Oct-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   (𝐺 ∈ USGraph β†’ ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2) ↔ (𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))))
 
Theoremusgr2pth0 29019* In a simply graph, there is a path of length 2 iff there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   (𝐺 ∈ USGraph β†’ ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 2) ↔ (𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑧 ∧ (π‘ƒβ€˜2) = 𝑦) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑧} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑧, 𝑦})))))
 
Theorempthdlem1 29020* Lemma 1 for pthd 29023. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 9-Feb-2021.)
(πœ‘ β†’ 𝑃 ∈ Word V)    &   π‘… = ((β™―β€˜π‘ƒ) βˆ’ 1)    &   (πœ‘ β†’ βˆ€π‘– ∈ (0..^(β™―β€˜π‘ƒ))βˆ€π‘— ∈ (1..^𝑅)(𝑖 β‰  𝑗 β†’ (π‘ƒβ€˜π‘–) β‰  (π‘ƒβ€˜π‘—)))    β‡’   (πœ‘ β†’ Fun β—‘(𝑃 β†Ύ (1..^𝑅)))
 
Theorempthdlem2lem 29021* Lemma for pthdlem2 29022. (Contributed by AV, 10-Feb-2021.)
(πœ‘ β†’ 𝑃 ∈ Word V)    &   π‘… = ((β™―β€˜π‘ƒ) βˆ’ 1)    &   (πœ‘ β†’ βˆ€π‘– ∈ (0..^(β™―β€˜π‘ƒ))βˆ€π‘— ∈ (1..^𝑅)(𝑖 β‰  𝑗 β†’ (π‘ƒβ€˜π‘–) β‰  (π‘ƒβ€˜π‘—)))    β‡’   ((πœ‘ ∧ (β™―β€˜π‘ƒ) ∈ β„• ∧ (𝐼 = 0 ∨ 𝐼 = 𝑅)) β†’ (π‘ƒβ€˜πΌ) βˆ‰ (𝑃 β€œ (1..^𝑅)))
 
Theorempthdlem2 29022* Lemma 2 for pthd 29023. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 10-Feb-2021.)
(πœ‘ β†’ 𝑃 ∈ Word V)    &   π‘… = ((β™―β€˜π‘ƒ) βˆ’ 1)    &   (πœ‘ β†’ βˆ€π‘– ∈ (0..^(β™―β€˜π‘ƒ))βˆ€π‘— ∈ (1..^𝑅)(𝑖 β‰  𝑗 β†’ (π‘ƒβ€˜π‘–) β‰  (π‘ƒβ€˜π‘—)))    β‡’   (πœ‘ β†’ ((𝑃 β€œ {0, 𝑅}) ∩ (𝑃 β€œ (1..^𝑅))) = βˆ…)
 
Theorempthd 29023* Two words representing a trail which also represent a path in a graph. (Contributed by AV, 10-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
(πœ‘ β†’ 𝑃 ∈ Word V)    &   π‘… = ((β™―β€˜π‘ƒ) βˆ’ 1)    &   (πœ‘ β†’ βˆ€π‘– ∈ (0..^(β™―β€˜π‘ƒ))βˆ€π‘— ∈ (1..^𝑅)(𝑖 β‰  𝑗 β†’ (π‘ƒβ€˜π‘–) β‰  (π‘ƒβ€˜π‘—)))    &   (β™―β€˜πΉ) = 𝑅    &   (πœ‘ β†’ 𝐹(Trailsβ€˜πΊ)𝑃)    β‡’   (πœ‘ β†’ 𝐹(Pathsβ€˜πΊ)𝑃)
 
17.3.5  Closed walks
 
Syntaxcclwlks 29024 Extend class notation with closed walks (of a graph).
class ClWalks
 
Definitiondf-clwlks 29025* 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 can be considered as a closed walk of length 0, see also 0clwlk 29380. (Contributed by Alexander van der Vekens, 12-Mar-2018.) (Revised by AV, 16-Feb-2021.)

ClWalks = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜π‘”)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)))})
 
Theoremclwlks 29026* The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) (Revised by AV, 29-Oct-2021.)
(ClWalksβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Walksβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)))}
 
Theoremisclwlk 29027 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.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(ClWalksβ€˜πΊ)𝑃 ↔ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
 
Theoremclwlkiswlk 29028 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.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(ClWalksβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
 
Theoremclwlkwlk 29029 Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
(π‘Š ∈ (ClWalksβ€˜πΊ) β†’ π‘Š ∈ (Walksβ€˜πΊ))
 
Theoremclwlkswks 29030 Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-Feb-2021.)
(ClWalksβ€˜πΊ) βŠ† (Walksβ€˜πΊ)
 
Theoremisclwlke 29031* 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.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   (𝐺 ∈ 𝑋 β†’ (𝐹(ClWalksβ€˜πΊ)𝑃 ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), (πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘˜))) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))))
 
Theoremisclwlkupgr 29032* Properties of a pair of functions to be a closed walk (in a pseudograph). (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 11-Apr-2021.) (Revised by AV, 28-Oct-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   (𝐺 ∈ UPGraph β†’ (𝐹(ClWalksβ€˜πΊ)𝑃 ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))))
 
Theoremclwlkcomp 29033* A closed walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   πΉ = (1st β€˜π‘Š)    &   π‘ƒ = (2nd β€˜π‘Š)    β‡’   ((𝐺 ∈ 𝑋 ∧ π‘Š ∈ (𝑆 Γ— 𝑇)) β†’ (π‘Š ∈ (ClWalksβ€˜πΊ) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), (πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘˜))) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))))
 
Theoremclwlkcompim 29034* 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.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   πΉ = (1st β€˜π‘Š)    &   π‘ƒ = (2nd β€˜π‘Š)    β‡’   (π‘Š ∈ (ClWalksβ€˜πΊ) β†’ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), (πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘˜))) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))))
 
Theoremupgrclwlkcompim 29035* Implications for the properties of the components of a closed walk in a pseudograph. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 2-May-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   πΉ = (1st β€˜π‘Š)    &   π‘ƒ = (2nd β€˜π‘Š)    β‡’   ((𝐺 ∈ UPGraph ∧ π‘Š ∈ (ClWalksβ€˜πΊ)) β†’ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
 
Theoremclwlkcompbp 29036 Basic properties of the components of a closed walk. (Contributed by AV, 23-May-2022.)
𝐹 = (1st β€˜π‘Š)    &   π‘ƒ = (2nd β€˜π‘Š)    β‡’   (π‘Š ∈ (ClWalksβ€˜πΊ) β†’ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
 
Theoremclwlkl1loop 29037 A closed walk of length 1 is a loop. (Contributed by AV, 22-Apr-2021.)
((Fun (iEdgβ€˜πΊ) ∧ 𝐹(ClWalksβ€˜πΊ)𝑃 ∧ (β™―β€˜πΉ) = 1) β†’ ((π‘ƒβ€˜0) = (π‘ƒβ€˜1) ∧ {(π‘ƒβ€˜0)} ∈ (Edgβ€˜πΊ)))
 
17.3.6  Circuits and cycles
 
Syntaxccrcts 29038 Extend class notation with circuits (in a graph).
class Circuits
 
Syntaxccycls 29039 Extend class notation with cycles (in a graph).
class Cycles
 
Definitiondf-crcts 29040* 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 = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜π‘”)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)))})
 
Definitiondf-cycls 29041* 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 = (𝑔 ∈ V ↦ {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Pathsβ€˜π‘”)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)))})
 
Theoremcrcts 29042* The set of circuits (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(Circuitsβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Trailsβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)))}
 
Theoremcycls 29043* The set of cycles (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(Cyclesβ€˜πΊ) = {βŸ¨π‘“, π‘βŸ© ∣ (𝑓(Pathsβ€˜πΊ)𝑝 ∧ (π‘β€˜0) = (π‘β€˜(β™―β€˜π‘“)))}
 
Theoremiscrct 29044 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.) (Revised by AV, 30-Oct-2021.)
(𝐹(Circuitsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
 
Theoremiscycl 29045 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.) (Revised by AV, 30-Oct-2021.)
(𝐹(Cyclesβ€˜πΊ)𝑃 ↔ (𝐹(Pathsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
 
Theoremcrctprop 29046 The properties of a circuit: A circuit is a closed trail. (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Circuitsβ€˜πΊ)𝑃 β†’ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
 
Theoremcyclprop 29047 The properties of a cycle: A cycle is a closed path. (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Cyclesβ€˜πΊ)𝑃 β†’ (𝐹(Pathsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
 
Theoremcrctisclwlk 29048 A circuit is a closed walk. (Contributed by AV, 17-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Circuitsβ€˜πΊ)𝑃 β†’ 𝐹(ClWalksβ€˜πΊ)𝑃)
 
Theoremcrctistrl 29049 A circuit is a trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(Circuitsβ€˜πΊ)𝑃 β†’ 𝐹(Trailsβ€˜πΊ)𝑃)
 
Theoremcrctiswlk 29050 A circuit is a walk. (Contributed by AV, 6-Apr-2021.)
(𝐹(Circuitsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
 
Theoremcyclispth 29051 A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(Cyclesβ€˜πΊ)𝑃 β†’ 𝐹(Pathsβ€˜πΊ)𝑃)
 
Theoremcycliswlk 29052 A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(Cyclesβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
 
Theoremcycliscrct 29053 A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Cyclesβ€˜πΊ)𝑃 β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)
 
Theoremcyclnspth 29054 A (non-trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹 β‰  βˆ… β†’ (𝐹(Cyclesβ€˜πΊ)𝑃 β†’ Β¬ 𝐹(SPathsβ€˜πΊ)𝑃))
 
Theoremcyclispthon 29055 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.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Cyclesβ€˜πΊ)𝑃 β†’ 𝐹((π‘ƒβ€˜0)(PathsOnβ€˜πΊ)(π‘ƒβ€˜0))𝑃)
 
Theoremlfgrn1cycl 29056* 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.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   (𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} β†’ (𝐹(Cyclesβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) β‰  1))
 
Theoremusgr2trlncrct 29057 In a simple graph, any trail of length 2 is not a circuit. (Contributed by AV, 5-Jun-2021.)
((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝐹(Trailsβ€˜πΊ)𝑃 β†’ Β¬ 𝐹(Circuitsβ€˜πΊ)𝑃))
 
Theoremumgrn1cycl 29058 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.)
((𝐺 ∈ UMGraph ∧ 𝐹(Cyclesβ€˜πΊ)𝑃) β†’ (β™―β€˜πΉ) β‰  1)
 
Theoremuspgrn2crct 29059 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.) (Proof shortened by AV, 31-Oct-2021.)
((𝐺 ∈ USPGraph ∧ 𝐹(Circuitsβ€˜πΊ)𝑃) β†’ (β™―β€˜πΉ) β‰  2)
 
Theoremusgrn2cycl 29060 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.)
((𝐺 ∈ USGraph ∧ 𝐹(Cyclesβ€˜πΊ)𝑃) β†’ (β™―β€˜πΉ) β‰  2)
 
Theoremcrctcshwlkn0lem1 29061 Lemma for crctcshwlkn0 29072. (Contributed by AV, 13-Mar-2021.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ β„•) β†’ ((𝐴 βˆ’ 𝐡) + 1) ≀ 𝐴)
 
Theoremcrctcshwlkn0lem2 29062* Lemma for crctcshwlkn0 29072. (Contributed by AV, 12-Mar-2021.)
(πœ‘ β†’ 𝑆 ∈ (1..^𝑁))    &   π‘„ = (π‘₯ ∈ (0...𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝑆), (π‘ƒβ€˜(π‘₯ + 𝑆)), (π‘ƒβ€˜((π‘₯ + 𝑆) βˆ’ 𝑁))))    β‡’   ((πœ‘ ∧ 𝐽 ∈ (0...(𝑁 βˆ’ 𝑆))) β†’ (π‘„β€˜π½) = (π‘ƒβ€˜(𝐽 + 𝑆)))
 
Theoremcrctcshwlkn0lem3 29063* Lemma for crctcshwlkn0 29072. (Contributed by AV, 12-Mar-2021.)
(πœ‘ β†’ 𝑆 ∈ (1..^𝑁))    &   π‘„ = (π‘₯ ∈ (0...𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝑆), (π‘ƒβ€˜(π‘₯ + 𝑆)), (π‘ƒβ€˜((π‘₯ + 𝑆) βˆ’ 𝑁))))    β‡’   ((πœ‘ ∧ 𝐽 ∈ (((𝑁 βˆ’ 𝑆) + 1)...𝑁)) β†’ (π‘„β€˜π½) = (π‘ƒβ€˜((𝐽 + 𝑆) βˆ’ 𝑁)))
 
Theoremcrctcshwlkn0lem4 29064* Lemma for crctcshwlkn0 29072. (Contributed by AV, 12-Mar-2021.)
(πœ‘ β†’ 𝑆 ∈ (1..^𝑁))    &   π‘„ = (π‘₯ ∈ (0...𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝑆), (π‘ƒβ€˜(π‘₯ + 𝑆)), (π‘ƒβ€˜((π‘₯ + 𝑆) βˆ’ 𝑁))))    &   π» = (𝐹 cyclShift 𝑆)    &   π‘ = (β™―β€˜πΉ)    &   (πœ‘ β†’ 𝐹 ∈ Word 𝐴)    &   (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝑁)if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))))    β‡’   (πœ‘ β†’ βˆ€π‘— ∈ (0..^(𝑁 βˆ’ 𝑆))if-((π‘„β€˜π‘—) = (π‘„β€˜(𝑗 + 1)), (πΌβ€˜(π»β€˜π‘—)) = {(π‘„β€˜π‘—)}, {(π‘„β€˜π‘—), (π‘„β€˜(𝑗 + 1))} βŠ† (πΌβ€˜(π»β€˜π‘—))))
 
Theoremcrctcshwlkn0lem5 29065* Lemma for crctcshwlkn0 29072. (Contributed by AV, 12-Mar-2021.)
(πœ‘ β†’ 𝑆 ∈ (1..^𝑁))    &   π‘„ = (π‘₯ ∈ (0...𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝑆), (π‘ƒβ€˜(π‘₯ + 𝑆)), (π‘ƒβ€˜((π‘₯ + 𝑆) βˆ’ 𝑁))))    &   π» = (𝐹 cyclShift 𝑆)    &   π‘ = (β™―β€˜πΉ)    &   (πœ‘ β†’ 𝐹 ∈ Word 𝐴)    &   (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝑁)if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))))    β‡’   (πœ‘ β†’ βˆ€π‘— ∈ (((𝑁 βˆ’ 𝑆) + 1)..^𝑁)if-((π‘„β€˜π‘—) = (π‘„β€˜(𝑗 + 1)), (πΌβ€˜(π»β€˜π‘—)) = {(π‘„β€˜π‘—)}, {(π‘„β€˜π‘—), (π‘„β€˜(𝑗 + 1))} βŠ† (πΌβ€˜(π»β€˜π‘—))))
 
Theoremcrctcshwlkn0lem6 29066* Lemma for crctcshwlkn0 29072. (Contributed by AV, 12-Mar-2021.)
(πœ‘ β†’ 𝑆 ∈ (1..^𝑁))    &   π‘„ = (π‘₯ ∈ (0...𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝑆), (π‘ƒβ€˜(π‘₯ + 𝑆)), (π‘ƒβ€˜((π‘₯ + 𝑆) βˆ’ 𝑁))))    &   π» = (𝐹 cyclShift 𝑆)    &   π‘ = (β™―β€˜πΉ)    &   (πœ‘ β†’ 𝐹 ∈ Word 𝐴)    &   (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝑁)if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))))    &   (πœ‘ β†’ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜0))    β‡’   ((πœ‘ ∧ 𝐽 = (𝑁 βˆ’ 𝑆)) β†’ if-((π‘„β€˜π½) = (π‘„β€˜(𝐽 + 1)), (πΌβ€˜(π»β€˜π½)) = {(π‘„β€˜π½)}, {(π‘„β€˜π½), (π‘„β€˜(𝐽 + 1))} βŠ† (πΌβ€˜(π»β€˜π½))))
 
Theoremcrctcshwlkn0lem7 29067* Lemma for crctcshwlkn0 29072. (Contributed by AV, 12-Mar-2021.)
(πœ‘ β†’ 𝑆 ∈ (1..^𝑁))    &   π‘„ = (π‘₯ ∈ (0...𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝑆), (π‘ƒβ€˜(π‘₯ + 𝑆)), (π‘ƒβ€˜((π‘₯ + 𝑆) βˆ’ 𝑁))))    &   π» = (𝐹 cyclShift 𝑆)    &   π‘ = (β™―β€˜πΉ)    &   (πœ‘ β†’ 𝐹 ∈ Word 𝐴)    &   (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝑁)if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))))    &   (πœ‘ β†’ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜0))    β‡’   (πœ‘ β†’ βˆ€π‘— ∈ (0..^𝑁)if-((π‘„β€˜π‘—) = (π‘„β€˜(𝑗 + 1)), (πΌβ€˜(π»β€˜π‘—)) = {(π‘„β€˜π‘—)}, {(π‘„β€˜π‘—), (π‘„β€˜(𝑗 + 1))} βŠ† (πΌβ€˜(π»β€˜π‘—))))
 
Theoremcrctcshlem1 29068 Lemma for crctcsh 29075. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   (πœ‘ β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)    &   π‘ = (β™―β€˜πΉ)    β‡’   (πœ‘ β†’ 𝑁 ∈ β„•0)
 
Theoremcrctcshlem2 29069 Lemma for crctcsh 29075. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   (πœ‘ β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)    &   π‘ = (β™―β€˜πΉ)    &   (πœ‘ β†’ 𝑆 ∈ (0..^𝑁))    &   π» = (𝐹 cyclShift 𝑆)    β‡’   (πœ‘ β†’ (β™―β€˜π») = 𝑁)
 
Theoremcrctcshlem3 29070* Lemma for crctcsh 29075. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   (πœ‘ β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)    &   π‘ = (β™―β€˜πΉ)    &   (πœ‘ β†’ 𝑆 ∈ (0..^𝑁))    &   π» = (𝐹 cyclShift 𝑆)    &   π‘„ = (π‘₯ ∈ (0...𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝑆), (π‘ƒβ€˜(π‘₯ + 𝑆)), (π‘ƒβ€˜((π‘₯ + 𝑆) βˆ’ 𝑁))))    β‡’   (πœ‘ β†’ (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
 
Theoremcrctcshlem4 29071* Lemma for crctcsh 29075. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   (πœ‘ β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)    &   π‘ = (β™―β€˜πΉ)    &   (πœ‘ β†’ 𝑆 ∈ (0..^𝑁))    &   π» = (𝐹 cyclShift 𝑆)    &   π‘„ = (π‘₯ ∈ (0...𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝑆), (π‘ƒβ€˜(π‘₯ + 𝑆)), (π‘ƒβ€˜((π‘₯ + 𝑆) βˆ’ 𝑁))))    β‡’   ((πœ‘ ∧ 𝑆 = 0) β†’ (𝐻 = 𝐹 ∧ 𝑄 = 𝑃))
 
Theoremcrctcshwlkn0 29072* Cyclically shifting the indices of a circuit ⟨𝐹, π‘ƒβŸ© results in a walk ⟨𝐻, π‘„βŸ©. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   (πœ‘ β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)    &   π‘ = (β™―β€˜πΉ)    &   (πœ‘ β†’ 𝑆 ∈ (0..^𝑁))    &   π» = (𝐹 cyclShift 𝑆)    &   π‘„ = (π‘₯ ∈ (0...𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝑆), (π‘ƒβ€˜(π‘₯ + 𝑆)), (π‘ƒβ€˜((π‘₯ + 𝑆) βˆ’ 𝑁))))    β‡’   ((πœ‘ ∧ 𝑆 β‰  0) β†’ 𝐻(Walksβ€˜πΊ)𝑄)
 
Theoremcrctcshwlk 29073* Cyclically shifting the indices of a circuit ⟨𝐹, π‘ƒβŸ© results in a walk ⟨𝐻, π‘„βŸ©. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   (πœ‘ β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)    &   π‘ = (β™―β€˜πΉ)    &   (πœ‘ β†’ 𝑆 ∈ (0..^𝑁))    &   π» = (𝐹 cyclShift 𝑆)    &   π‘„ = (π‘₯ ∈ (0...𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝑆), (π‘ƒβ€˜(π‘₯ + 𝑆)), (π‘ƒβ€˜((π‘₯ + 𝑆) βˆ’ 𝑁))))    β‡’   (πœ‘ β†’ 𝐻(Walksβ€˜πΊ)𝑄)
 
Theoremcrctcshtrl 29074* Cyclically shifting the indices of a circuit ⟨𝐹, π‘ƒβŸ© results in a trail ⟨𝐻, π‘„βŸ©. (Contributed by AV, 14-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   (πœ‘ β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)    &   π‘ = (β™―β€˜πΉ)    &   (πœ‘ β†’ 𝑆 ∈ (0..^𝑁))    &   π» = (𝐹 cyclShift 𝑆)    &   π‘„ = (π‘₯ ∈ (0...𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝑆), (π‘ƒβ€˜(π‘₯ + 𝑆)), (π‘ƒβ€˜((π‘₯ + 𝑆) βˆ’ 𝑁))))    β‡’   (πœ‘ β†’ 𝐻(Trailsβ€˜πΊ)𝑄)
 
Theoremcrctcsh 29075* Cyclically shifting the indices of a circuit ⟨𝐹, π‘ƒβŸ© results in a circuit ⟨𝐻, π‘„βŸ©. (Contributed by AV, 10-Mar-2021.) (Proof shortened by AV, 31-Oct-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   (πœ‘ β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)    &   π‘ = (β™―β€˜πΉ)    &   (πœ‘ β†’ 𝑆 ∈ (0..^𝑁))    &   π» = (𝐹 cyclShift 𝑆)    &   π‘„ = (π‘₯ ∈ (0...𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝑆), (π‘ƒβ€˜(π‘₯ + 𝑆)), (π‘ƒβ€˜((π‘₯ + 𝑆) βˆ’ 𝑁))))    β‡’   (πœ‘ β†’ 𝐻(Circuitsβ€˜πΊ)𝑄)
 
17.3.7  Walks as words

In general, a walk is an alternating sequence of vertices and edges, as defined in df-wlks 28853: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Often, it is sufficient to refer to a walk by the natural sequence of its vertices, i.e omitting its edges in its representation: p(0) p(1) ... p(n-1) p(n), see the corresponding remark in [Diestel] p. 6. The concept of a Word, see df-word 14464, is the appropriate way to define such a sequence (being finite and starting at index 0) of vertices. Therefore, it is used in Definitions df-wwlks 29081 and df-wwlksn 29082, and the representation of a walk as sequence of its vertices is called "walk as word".

Only for simple pseudographs, however, the edges can be uniquely reconstructed from such a representation. In other cases, there could be more than one edge between two adjacent vertices in the walk (in a multigraph), or two adjacent vertices could be connected by two different hyperedges involving additional vertices (in a hypergraph).

 
Syntaxcwwlks 29076 Extend class notation with walks (in a graph) as word over the set of vertices.
class WWalks
 
Syntaxcwwlksn 29077 Extend class notation with walks (in a graph) of a fixed length as word over the set of vertices.
class WWalksN
 
Syntaxcwwlksnon 29078 Extend class notation with walks between two vertices (in a graph) of a fixed length as word over the set of vertices.
class WWalksNOn
 
Syntaxcwwspthsn 29079 Extend class notation with simple paths (in a graph) of a fixed length as word over the set of vertices.
class WSPathsN
 
Syntaxcwwspthsnon 29080 Extend class notation with simple paths between two vertices (in a graph) of a fixed length as word over the set of vertices.
class WSPathsNOn
 
Definitiondf-wwlks 29081* Define the set of all walks (in an undirected graph) as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n-1) p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlks 28853. 𝑀 = βˆ… has to be excluded because a walk always consists of at least one vertex, see wlkn0 28875. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
WWalks = (𝑔 ∈ V ↦ {𝑀 ∈ Word (Vtxβ€˜π‘”) ∣ (𝑀 β‰  βˆ… ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ (Edgβ€˜π‘”))})
 
Definitiondf-wwlksn 29082* Define the set of all walks (in an undirected graph) of a fixed length n as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlks 28853. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
WWalksN = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ {𝑀 ∈ (WWalksβ€˜π‘”) ∣ (β™―β€˜π‘€) = (𝑛 + 1)})
 
Definitiondf-wwlksnon 29083* Define the collection of walks of a fixed length with particular endpoints as word over the set of vertices. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 11-May-2021.)
WWalksNOn = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (𝑛 WWalksN 𝑔) ∣ ((π‘€β€˜0) = π‘Ž ∧ (π‘€β€˜π‘›) = 𝑏)}))
 
Definitiondf-wspthsn 29084* Define the collection of simple paths of a fixed length as word over the set of vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
WSPathsN = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ {𝑀 ∈ (𝑛 WWalksN 𝑔) ∣ βˆƒπ‘“ 𝑓(SPathsβ€˜π‘”)𝑀})
 
Definitiondf-wspthsnon 29085* Define the collection of simple paths of a fixed length with particular endpoints as word over the set of vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
WSPathsNOn = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀}))
 
Theoremwwlks 29086* The set of walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΈ = (Edgβ€˜πΊ)    β‡’   (WWalksβ€˜πΊ) = {𝑀 ∈ Word 𝑉 ∣ (𝑀 β‰  βˆ… ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘€) βˆ’ 1)){(π‘€β€˜π‘–), (π‘€β€˜(𝑖 + 1))} ∈ 𝐸)}
 
Theoremiswwlks 29087* A word over the set of vertices representing a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΈ = (Edgβ€˜πΊ)    β‡’   (π‘Š ∈ (WWalksβ€˜πΊ) ↔ (π‘Š β‰  βˆ… ∧ π‘Š ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸))
 
Theoremwwlksn 29088* The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
(𝑁 ∈ β„•0 β†’ (𝑁 WWalksN 𝐺) = {𝑀 ∈ (WWalksβ€˜πΊ) ∣ (β™―β€˜π‘€) = (𝑁 + 1)})
 
Theoremiswwlksn 29089 A word over the set of vertices representing a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
(𝑁 ∈ β„•0 β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) ↔ (π‘Š ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1))))
 
Theoremwwlksnprcl 29090 Derivation of the length of a word π‘Š whose concatenation with a singleton word represents a walk of a fixed length 𝑁 (a partial reverse closure theorem). (Contributed by AV, 4-Mar-2022.)
((π‘Š ∈ Word 𝑉 ∧ 𝑁 ∈ β„•0) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ∈ (𝑁 WWalksN 𝐺) β†’ (β™―β€˜π‘Š) = 𝑁))
 
Theoremiswwlksnx 29091* Properties of a word to represent a walk of a fixed length, definition of WWalks expanded. (Contributed by AV, 28-Apr-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΈ = (Edgβ€˜πΊ)    β‡’   (𝑁 ∈ β„•0 β†’ (π‘Š ∈ (𝑁 WWalksN 𝐺) ↔ (π‘Š ∈ Word 𝑉 ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸 ∧ (β™―β€˜π‘Š) = (𝑁 + 1))))
 
Theoremwwlkbp 29092 Basic properties of a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (π‘Š ∈ (WWalksβ€˜πΊ) β†’ (𝐺 ∈ V ∧ π‘Š ∈ Word 𝑉))
 
Theoremwwlknbp 29093 Basic properties of a walk of a fixed length (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 16-Jul-2018.) (Revised by AV, 9-Apr-2021.) (Proof shortened by AV, 20-May-2021.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ π‘Š ∈ Word 𝑉))
 
Theoremwwlknp 29094* Properties of a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   πΈ = (Edgβ€˜πΊ)    β‡’   (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 + 1) ∧ βˆ€π‘– ∈ (0..^𝑁){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ 𝐸))
 
Theoremwwlknbp1 29095 Other basic properties of a walk of a fixed length as word. (Contributed by AV, 8-Mar-2022.)
(π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)))
 
Theoremwwlknvtx 29096* The symbols of a word π‘Š representing a walk of a fixed length 𝑁 are vertices. (Contributed by AV, 16-Mar-2022.)
(π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ βˆ€π‘– ∈ (0...𝑁)(π‘Šβ€˜π‘–) ∈ (Vtxβ€˜πΊ))
 
Theoremwwlknllvtx 29097 If a word π‘Š represents a walk of a fixed length 𝑁, then the first and the last symbol of the word is a vertex. (Contributed by AV, 14-Mar-2022.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ ((π‘Šβ€˜0) ∈ 𝑉 ∧ (π‘Šβ€˜π‘) ∈ 𝑉))
 
Theoremwwlknlsw 29098 If a word represents a walk of a fixed length, then the last symbol of the word is the endvertex of the walk. (Contributed by AV, 8-Mar-2022.)
(π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (π‘Šβ€˜π‘) = (lastSβ€˜π‘Š))
 
Theoremwspthsn 29099* The set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
(𝑁 WSPathsN 𝐺) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)𝑀}
 
Theoremiswspthn 29100* An element of the set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
(π‘Š ∈ (𝑁 WSPathsN 𝐺) ↔ (π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ βˆƒπ‘“ 𝑓(SPathsβ€˜πΊ)π‘Š))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 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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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