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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremclwwlknon1nloop 27501 If there is no loop at vertex 𝑋, the set of (closed) walks on 𝑋 of length 1 as words over the set of vertices is empty. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Mar-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐶 = (ClWWalksNOn‘𝐺)    &   𝐸 = (Edg‘𝐺)       ({𝑋} ∉ 𝐸 → (𝑋𝐶1) = ∅)

Theoremclwwlknon1sn 27502 The set of (closed) walks on vertex 𝑋 of length 1 as words over the set of vertices is a singleton containing the singleton word consisting of 𝑋 iff there is a loop at 𝑋. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐶 = (ClWWalksNOn‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑋𝑉 → ((𝑋𝐶1) = {⟨“𝑋”⟩} ↔ {𝑋} ∈ 𝐸))

Theoremclwwlknon1le1 27503 There is at most one (closed) walk on vertex 𝑋 of length 1 as word over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Mar-2022.)
(♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1

Theoremclwwlknon2 27504* The set of closed walks on vertex 𝑋 of length 2 in a graph 𝐺 as words over the set of vertices. (Contributed by AV, 5-Mar-2022.) (Proof revised by AV, 25-Mar-2022.)
𝐶 = (ClWWalksNOn‘𝐺)       (𝑋𝐶2) = {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}

Theoremclwwlknon2x 27505* The set of closed walks on vertex 𝑋 of length 2 in a graph 𝐺 as words over the set of vertices, definition of ClWWalksN expanded. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Mar-2022.)
𝐶 = (ClWWalksNOn‘𝐺)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑋𝐶2) = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)}

Theorems2elclwwlknon2 27506 Sufficient conditions of a doubleton word to represent a closed walk on vertex 𝑋 of length 2. (Contributed by AV, 11-May-2022.)
𝐶 = (ClWWalksNOn‘𝐺)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝑋𝑉𝑌𝑉 ∧ {𝑋, 𝑌} ∈ 𝐸) → ⟨“𝑋𝑌”⟩ ∈ (𝑋𝐶2))

Theoremclwwlknon2num 27507 In a 𝐾-regular graph 𝐺, there are 𝐾 closed walks on vertex 𝑋 of length 2. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 25-Mar-2022.)
((𝐺RegUSGraph𝐾𝑋 ∈ (Vtx‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)

Theoremclwwlknonwwlknonb 27508 A word over vertices represents a closed walk of a fixed length 𝑁 on vertex 𝑋 iff the word concatenated with 𝑋 represents a walk of length 𝑁 on 𝑋 and 𝑋. This theorem would not hold for 𝑁 = 0 and 𝑊 = ∅, see clwwlknwwlksnb 27452. (Contributed by AV, 4-Mar-2022.) (Revised by AV, 27-Mar-2022.)
𝑉 = (Vtx‘𝐺)       ((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑋(𝑁 WWalksNOn 𝐺)𝑋)))

Theoremclwwlknonex2lem1 27509 Lemma 1 for clwwlknonex2 27511: Transformation of a special half-open integer range into a union of a smaller half-open integer range and an unordered pair. This Lemma would not hold for 𝑁 = 2, i.e., (♯‘𝑊) = 0, because (0..^(((♯‘𝑊) + 2) − 1)) = (0..^((0 + 2) − 1)) = (0..^1) = {0} ≠ {-1, 0} = (∅ ∪ {-1, 0}) = ((0..^(0 − 1)) ∪ {(0 − 1), 0}) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (♯‘𝑊)}). (Contributed by AV, 22-Sep-2018.) (Revised by AV, 26-Jan-2022.)
((𝑁 ∈ (ℤ‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → (0..^(((♯‘𝑊) + 2) − 1)) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (♯‘𝑊)}))

Theoremclwwlknonex2lem2 27510* Lemma 2 for clwwlknonex2 27511: Transformation of a walk and two edges into a walk extended by two vertices/edges. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 27-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((((𝑋𝑉𝑌𝑉𝑁 ∈ (ℤ‘3)) ∧ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝑊‘0) = 𝑋)) ∧ {𝑋, 𝑌} ∈ 𝐸) → ∀𝑖 ∈ ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (♯‘𝑊)}){(((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝑖), (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘(𝑖 + 1))} ∈ 𝐸)

Theoremclwwlknonex2 27511 Extending a closed walk 𝑊 on vertex 𝑋 by an additional edge (forth and back) results in a closed walk. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 28-Mar-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝑋𝑉𝑌𝑉𝑁 ∈ (ℤ‘3)) ∧ {𝑋, 𝑌} ∈ 𝐸𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ (𝑁 ClWWalksN 𝐺))

Theoremclwwlknonex2e 27512 Extending a closed walk 𝑊 on vertex 𝑋 by an additional edge (forth and back) results in a closed walk on vertex 𝑋. (Contributed by AV, 17-Apr-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝑋𝑉𝑌𝑉𝑁 ∈ (ℤ‘3)) ∧ {𝑋, 𝑌} ∈ 𝐸𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁))

Theoremclwwlknondisj 27513* The sets of closed walks on different vertices are disjunct. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)
Disj 𝑥𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁)

Theoremclwwlknun 27514* The set of closed walks of fixed length 𝑁 in a simple graph 𝐺 is the union of the closed walks of the fixed length 𝑁 on each of the vertices of graph 𝐺. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ USGraph → (𝑁 ClWWalksN 𝐺) = 𝑥𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁))

Theoremclwwlkvbij 27515* There is a bijection between the set of closed walks of a fixed length 𝑁 on a fixed vertex 𝑋 represented by walks (as word) and the set of closed walks (as words) of the fixed length 𝑁 on the fixed vertex 𝑋. The difference between these two representations is that in the first case the fixed vertex is repeated at the end of the word, and in the second case it is not. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 7-Jul-2022.) (Proof shortened by AV, 2-Nov-2022.)
((𝑋𝑉𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))

TheoremclwwlkvbijOLD 27516* Obsolete proof of clwwlkvbij 27515 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 7-Jul-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑋𝑉𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))

16.3.11  Examples for walks, trails and paths

Theorem0ewlk 27517 The empty set (empty sequence of edges) is an s-walk of edges for all s. (Contributed by AV, 4-Jan-2021.)
((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) → ∅ ∈ (𝐺 EdgWalks 𝑆))

Theorem1ewlk 27518 A sequence of 1 edge is an s-walk of edges for all s. (Contributed by AV, 5-Jan-2021.)
((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*𝐼 ∈ dom (iEdg‘𝐺)) → ⟨“𝐼”⟩ ∈ (𝐺 EdgWalks 𝑆))

Theorem0wlk 27519 A pair of an empty set (of edges) and a second set (of vertices) is a walk iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑈 → (∅(Walks‘𝐺)𝑃𝑃:(0...0)⟶𝑉))

Theoremis0wlk 27520 A pair of an empty set (of edges) and a sequence of one vertex is a walk (of length 0). (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑃 = {⟨0, 𝑁⟩} ∧ 𝑁𝑉) → ∅(Walks‘𝐺)𝑃)

Theorem0wlkonlem1 27521 Lemma 1 for 0wlkon 27523 and 0trlon 27527. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → (𝑁𝑉𝑁𝑉))

Theorem0wlkonlem2 27522 Lemma 2 for 0wlkon 27523 and 0trlon 27527. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → 𝑃 ∈ (𝑉pm (0...0)))

Theorem0wlkon 27523 A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(WalksOn‘𝐺)𝑁)𝑃)

Theorem0wlkons1 27524 A walk of length 0 from a vertex to itself. (Contributed by AV, 17-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉 → ∅(𝑁(WalksOn‘𝐺)𝑁)⟨“𝑁”⟩)

Theorem0trl 27525 A pair of an empty set (of edges) and a second set (of vertices) is a trail iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑈 → (∅(Trails‘𝐺)𝑃𝑃:(0...0)⟶𝑉))

Theoremis0trl 27526 A pair of an empty set (of edges) and a sequence of one vertex is a trail (of length 0). (Contributed by AV, 7-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑃 = {⟨0, 𝑁⟩} ∧ 𝑁𝑉) → ∅(Trails‘𝐺)𝑃)

Theorem0trlon 27527 A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 8-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(TrailsOn‘𝐺)𝑁)𝑃)

Theorem0pth 27528 A pair of an empty set (of edges) and a second set (of vertices) is a path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 19-Jan-2021.) (Revised by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (∅(Paths‘𝐺)𝑃𝑃:(0...0)⟶𝑉))

Theorem0spth 27529 A pair of an empty set (of edges) and a second set (of vertices) is a simple path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 18-Jan-2021.) (Revised by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (∅(SPaths‘𝐺)𝑃𝑃:(0...0)⟶𝑉))

Theorem0pthon 27530 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.) (Revised by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(PathsOn‘𝐺)𝑁)𝑃)

Theorem0pthon1 27531 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉 → ∅(𝑁(PathsOn‘𝐺)𝑁){⟨0, 𝑁⟩})

Theorem0pthonv 27532* For each vertex there is a path of length 0 from the vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 21-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉 → ∃𝑓𝑝 𝑓(𝑁(PathsOn‘𝐺)𝑁)𝑝)

Theorem0clwlk 27533 A pair of an empty set (of edges) and a second set (of vertices) is a closed walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 17-Feb-2021.) (Revised by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑋 → (∅(ClWalks‘𝐺)𝑃𝑃:(0...0)⟶𝑉))

Theorem0clwlkv 27534 Any vertex (more precisely, a pair of an empty set (of edges) and a singleton function to this vertex) determines a closed walk of length 0. (Contributed by AV, 11-Feb-2022.)
𝑉 = (Vtx‘𝐺)       ((𝑋𝑉𝐹 = ∅ ∧ 𝑃:{0}⟶{𝑋}) → 𝐹(ClWalks‘𝐺)𝑃)

Theorem0clwlk0 27535 There is no closed walk in the empty set (i.e. the null graph). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
(ClWalks‘∅) = ∅

Theorem0crct 27536 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.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.)
(𝐺𝑊 → (∅(Circuits‘𝐺)𝑃𝑃:(0...0)⟶(Vtx‘𝐺)))

Theorem0cycl 27537 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.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.)
(𝐺𝑊 → (∅(Cycles‘𝐺)𝑃𝑃:(0...0)⟶(Vtx‘𝐺)))

Theorem1pthdlem1 27538 Lemma 1 for 1pthd 27546. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩       Fun (𝑃 ↾ (1..^(♯‘𝐹)))

Theorem1pthdlem2 27539 Lemma 2 for 1pthd 27546. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩       ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅

Theorem1wlkdlem1 27540 Lemma 1 for 1wlkd 27544. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑𝑃:(0...(♯‘𝐹))⟶𝑉)

Theorem1wlkdlem2 27541 Lemma 2 for 1wlkd 27544. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))       (𝜑𝑋 ∈ (𝐼𝐽))

Theorem1wlkdlem3 27542 Lemma 3 for 1wlkd 27544. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))       (𝜑𝐹 ∈ Word dom 𝐼)

Theorem1wlkdlem4 27543* Lemma 4 for 1wlkd 27544. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))       (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))

Theorem1wlkd 27544 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a walk. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(Walks‘𝐺)𝑃)

Theorem1trld 27545 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(Trails‘𝐺)𝑃)

Theorem1pthd 27546 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(Paths‘𝐺)𝑃)

Theorem1pthond 27547 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃)

Theoremupgr1wlkdlem1 27548 Lemma 1 for upgr1wlkd 27550. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋 ∈ (Vtx‘𝐺))    &   (𝜑𝑌 ∈ (Vtx‘𝐺))    &   (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})       ((𝜑𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋})

Theoremupgr1wlkdlem2 27549 Lemma 2 for upgr1wlkd 27550. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋 ∈ (Vtx‘𝐺))    &   (𝜑𝑌 ∈ (Vtx‘𝐺))    &   (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})       ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))

Theoremupgr1wlkd 27550 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a walk. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋 ∈ (Vtx‘𝐺))    &   (𝜑𝑌 ∈ (Vtx‘𝐺))    &   (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})    &   (𝜑𝐺 ∈ UPGraph)       (𝜑𝐹(Walks‘𝐺)𝑃)

Theoremupgr1trld 27551 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋 ∈ (Vtx‘𝐺))    &   (𝜑𝑌 ∈ (Vtx‘𝐺))    &   (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})    &   (𝜑𝐺 ∈ UPGraph)       (𝜑𝐹(Trails‘𝐺)𝑃)

Theoremupgr1pthd 27552 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋 ∈ (Vtx‘𝐺))    &   (𝜑𝑌 ∈ (Vtx‘𝐺))    &   (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})    &   (𝜑𝐺 ∈ UPGraph)       (𝜑𝐹(Paths‘𝐺)𝑃)

Theoremupgr1pthond 27553 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋 ∈ (Vtx‘𝐺))    &   (𝜑𝑌 ∈ (Vtx‘𝐺))    &   (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})    &   (𝜑𝐺 ∈ UPGraph)       (𝜑𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃)

Theoremlppthon 27554 A loop (which is an edge at index 𝐽) induces a path of length 1 from a vertex to itself in a hypergraph. (Contributed by AV, 1-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ⟨“𝐽”⟩(𝐴(PathsOn‘𝐺)𝐴)⟨“𝐴𝐴”⟩)

Theoremlp1cycl 27555 A loop (which is an edge at index 𝐽) induces a cycle of length 1 in a hypergraph. (Contributed by AV, 2-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ⟨“𝐽”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩)

Theorem1pthon2v 27556* For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)

Theorem1pthon2ve 27557* For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Proof shortened by AV, 15-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)

Theoremwlk2v2elem1 27558 Lemma 1 for wlk2v2e 27560: 𝐹 is a length 2 word of over {0}, the domain of the singleton word 𝐼. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.)
𝐼 = ⟨“{𝑋, 𝑌}”⟩    &   𝐹 = ⟨“00”⟩       𝐹 ∈ Word dom 𝐼

Theoremwlk2v2elem2 27559* Lemma 2 for wlk2v2e 27560: The values of 𝐼 after 𝐹 are edges between two vertices enumerated by 𝑃. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.)
𝐼 = ⟨“{𝑋, 𝑌}”⟩    &   𝐹 = ⟨“00”⟩    &   𝑋 ∈ V    &   𝑌 ∈ V    &   𝑃 = ⟨“𝑋𝑌𝑋”⟩       𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}

Theoremwlk2v2e 27560 In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk. Notice that 𝐺 is a simple graph (without loops) only if 𝑋𝑌. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
𝐼 = ⟨“{𝑋, 𝑌}”⟩    &   𝐹 = ⟨“00”⟩    &   𝑋 ∈ V    &   𝑌 ∈ V    &   𝑃 = ⟨“𝑋𝑌𝑋”⟩    &   𝐺 = ⟨{𝑋, 𝑌}, 𝐼       𝐹(Walks‘𝐺)𝑃

Theoremntrl2v2e 27561 A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk, see wlk2v2e 27560, but not a trail. Notice that 𝐺 is a simple graph (without loops) only if 𝑋𝑌. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝐼 = ⟨“{𝑋, 𝑌}”⟩    &   𝐹 = ⟨“00”⟩    &   𝑋 ∈ V    &   𝑌 ∈ V    &   𝑃 = ⟨“𝑋𝑌𝑋”⟩    &   𝐺 = ⟨{𝑋, 𝑌}, 𝐼        ¬ 𝐹(Trails‘𝐺)𝑃

Theorem3wlkdlem1 27562 Lemma 1 for 3wlkd 27573. (Contributed by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩       (♯‘𝑃) = ((♯‘𝐹) + 1)

Theorem3wlkdlem2 27563 Lemma 2 for 3wlkd 27573. (Contributed by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩       (0..^(♯‘𝐹)) = {0, 1, 2}

Theorem3wlkdlem3 27564 Lemma 3 for 3wlkd 27573. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))       (𝜑 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐷)))

Theorem3wlkdlem4 27565* Lemma 4 for 3wlkd 27573. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))       (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃𝑘) ∈ 𝑉)

Theorem3wlkdlem5 27566* Lemma 5 for 3wlkd 27573. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))       (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))

Theorem3pthdlem1 27567* Lemma 1 for 3pthd 27577. (Contributed by AV, 9-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))       (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘𝑗 → (𝑃𝑘) ≠ (𝑃𝑗)))

Theorem3wlkdlem6 27568 Lemma 6 for 3wlkd 27573. (Contributed by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))       (𝜑 → (𝐴 ∈ (𝐼𝐽) ∧ 𝐵 ∈ (𝐼𝐾) ∧ 𝐶 ∈ (𝐼𝐿)))

Theorem3wlkdlem7 27569 Lemma 7 for 3wlkd 27573. (Contributed by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))       (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V))

Theorem3wlkdlem8 27570 Lemma 8 for 3wlkd 27573. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))       (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿))

Theorem3wlkdlem9 27571 Lemma 9 for 3wlkd 27573. (Contributed by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))       (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ∧ {𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1)) ∧ {𝐶, 𝐷} ⊆ (𝐼‘(𝐹‘2))))

Theorem3wlkdlem10 27572* Lemma 10 for 3wlkd 27573. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))       (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))

Theorem3wlkd 27573 Construction of a walk from two given edges in a graph. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(Walks‘𝐺)𝑃)

Theorem3wlkond 27574 A walk of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(𝐴(WalksOn‘𝐺)𝐷)𝑃)

Theorem3trld 27575 Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))       (𝜑𝐹(Trails‘𝐺)𝑃)

Theorem3trlond 27576 A trail of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))       (𝜑𝐹(𝐴(TrailsOn‘𝐺)𝐷)𝑃)

Theorem3pthd 27577 A path of length 3 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))       (𝜑𝐹(Paths‘𝐺)𝑃)

Theorem3pthond 27578 A path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))       (𝜑𝐹(𝐴(PathsOn‘𝐺)𝐷)𝑃)

Theorem3spthd 27579 A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))    &   (𝜑𝐴𝐷)       (𝜑𝐹(SPaths‘𝐺)𝑃)

Theorem3spthond 27580 A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))    &   (𝜑𝐴𝐷)       (𝜑𝐹(𝐴(SPathsOn‘𝐺)𝐷)𝑃)

Theorem3cycld 27581 Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))    &   (𝜑𝐴 = 𝐷)       (𝜑𝐹(Cycles‘𝐺)𝑃)

Theorem3cyclpd 27582 Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))    &   (𝜑𝐴 = 𝐷)       (𝜑 → (𝐹(Cycles‘𝐺)𝑃 ∧ (♯‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴))

Theoremupgr3v3e3cycl 27583* If there is a cycle of length 3 in a pseudograph, there are three distinct vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ (♯‘𝐹) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐𝑐𝑎)))

Theoremuhgr3cyclexlem 27584 Lemma for uhgr3cyclex 27585. (Contributed by AV, 12-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((((𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → 𝐽𝐾)

Theoremuhgr3cyclex 27585* If there are three different vertices in a hypergraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ ((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))

Theoremumgr3cyclex 27586* If there are three (different) vertices in a multigraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))

Theoremumgr3v3e3cycl 27587* If and only if there is a 3-cycle in a multigraph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.) (Revised by AV, 12-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ UMGraph → (∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))

Theoremupgr4cycl4dv4e 27588* If there is a cycle of length 4 in a pseudograph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 13-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(Cycles‘𝐺)𝑃 ∧ (♯‘𝐹) = 4) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 ((({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ∧ ({𝑐, 𝑑} ∈ 𝐸 ∧ {𝑑, 𝑎} ∈ 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑))))

16.3.12  Connected graphs

Syntaxcconngr 27589 Extend class notation with connected graphs.
class ConnGraph

Definitiondf-conngr 27590* Define the class of all connected graphs. A graph is called connected if there is a path between every pair of (distinct) vertices. The distinctness of the vertices is not necessary for the definition, because there is always a path (of length 0) from a vertex to itself, see 0pthonv 27532 and dfconngr1 27591. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
ConnGraph = {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}

Theoremdfconngr1 27591* Alternative definition of the class of all connected graphs, requiring paths between distinct vertices. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 15-Feb-2021.)
ConnGraph = {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛 ∈ (𝑣 ∖ {𝑘})∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}

Theoremisconngr 27592* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))

Theoremisconngr1 27593* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))

Theoremcusconngr 27594 A complete hypergraph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 15-Feb-2021.)
((𝐺 ∈ UHGraph ∧ 𝐺 ∈ ComplGraph) → 𝐺 ∈ ConnGraph)

Theorem0conngr 27595 A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
∅ ∈ ConnGraph

Theorem0vconngr 27596 A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ ConnGraph)

Theorem1conngr 27597 A graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ ConnGraph)

Theoremconngrv2edg 27598* A vertex in a connected graph with more than one vertex is incident with at least one edge. Formerly part of proof for vdgn0frgrv2 27703. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ ConnGraph ∧ 𝑁𝑉 ∧ 1 < (♯‘𝑉)) → ∃𝑒 ∈ ran 𝐼 𝑁𝑒)

Theoremvdn0conngrumgrv2 27599 A vertex in a connected multigraph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁𝑉 ∧ 1 < (♯‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) ≠ 0)

16.4  Eulerian paths and the Konigsberg Bridge problem

16.4.1  Eulerian paths

According to Wikipedia ("Eulerian path", 9-Mar-2021, https://en.wikipedia.org/wiki/Eulerian_path): "In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. ... The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. ... A graph that has an Eulerian trail but not an Eulerian circuit is called semi-Eulerian."

Correspondingly, an Eulerian path is defined as "a trail containing all edges" (see definition in [Bollobas] p. 16) in df-eupth 27601 resp. iseupth 27604. (EulerPaths‘𝐺) is the set of all Eulerian paths in graph 𝐺, see eupths 27603. An Eulerian circuit (called Euler tour in the definition in [Diestel] p. 22) is "a circuit in a graph containing all the edges" (see definition in [Bollobas] p. 16), or, with other words, a circuit which is an Eulerian path. The function mapping a graph to the set of its Eulerian paths is defined as EulerPaths in df-eupth 27601, whereas there is no explicit definition for Eulerian circuits (yet): The statement "𝐹, 𝑃 is an Eulerian circuit" is formally expressed by (𝐹(EulerPaths‘𝐺)𝑃𝐹(Circuits‘𝐺)𝑃).

Each Eulerian path can be made an Eulerian circuit by adding an edge which connects the endpoints of the Eulerian path (see eupth2eucrct 27621). Vice versa, removing one edge from a graph with an Eulerian circuit results in a graph with an Eulerian path, see eucrct2eupth 27651.

An Eulerian path does not have to be a path in the meaning of definition df-pths 27068, because it may traverse some vertices more than once. Therefore, "Eulerian trail" would be a more appropriate name.

The main result of this section is (one direction of) Euler's Theorem: "A non-trivial connected graph has an Euler[ian] circuit iff each vertex has even degree." (see part 1 of theorem 12 in [Bollobas] p. 16 and theorem 1.8.1 in [Diestel] p. 22) or, expressed with Eulerian paths: "A connected graph has an Euler[ian] trail from a vertex x to a vertex y (not equal with x) iff x and y are the only vertices of odd degree." (see part 2 of theorem 12 in [Bollobas] p. 17). In eulerpath 27645, it is shown that a pseudograph with an Eulerian path has either zero or two vertices of odd degree, and eulercrct 27646 shows that a pseudograph with an Eulerian circuit has only vertices of even degree.

Syntaxceupth 27600 Extend class notation with Eulerian paths.
class EulerPaths

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 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 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43639
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