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Type | Label | Description |
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Statement | ||
Theorem | clwwlknonfin 27801 | In a finite graph 𝐺, the set of closed walks on vertex 𝑋 of length 𝑁 is also finite. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 24-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ Fin) | ||
Theorem | clwwlknonel 27802* | Characterization of a word over the set of vertices representing a closed walk on vertex 𝑋 of (nonzero) length 𝑁 in a graph 𝐺. This theorem would not hold for 𝑁 = 0 if 𝑊 = 𝑋 = ∅. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 24-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝑁 ≠ 0 → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (♯‘𝑊) = 𝑁 ∧ (𝑊‘0) = 𝑋))) | ||
Theorem | clwwlknonccat 27803 | The concatenation of two words representing closed walks on a vertex 𝑋 represents a closed walk on vertex 𝑋. The resulting walk is a "double loop", starting at vertex 𝑋, coming back to 𝑋 by the first walk, following the second walk and finally coming back to 𝑋 again. (Contributed by AV, 24-Apr-2022.) |
⊢ ((𝐴 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑀) ∧ 𝐵 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁)) → (𝐴 ++ 𝐵) ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑀 + 𝑁))) | ||
Theorem | clwwlknon1 27804* | The set of closed walks on vertex 𝑋 of length 1 in a graph 𝐺 as words over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 24-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐶 = (ClWWalksNOn‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = 〈“𝑋”〉 ∧ {𝑋} ∈ 𝐸)}) | ||
Theorem | clwwlknon1loop 27805 | If there is a loop at vertex 𝑋, the set of (closed) walks on 𝑋 of length 1 as words over the set of vertices is a singleton containing the singleton word consisting of 𝑋. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 25-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐶 = (ClWWalksNOn‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑋𝐶1) = {〈“𝑋”〉}) | ||
Theorem | clwwlknon1nloop 27806 | 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) = ∅) | ||
Theorem | clwwlknon1sn 27807 | 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) ⇒ ⊢ (𝑋 ∈ 𝑉 → ((𝑋𝐶1) = {〈“𝑋”〉} ↔ {𝑋} ∈ 𝐸)) | ||
Theorem | clwwlknon1le1 27808 | 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 | ||
Theorem | clwwlknon2 27809* | 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.) (Revised by AV, 25-Mar-2022.) |
⊢ 𝐶 = (ClWWalksNOn‘𝐺) ⇒ ⊢ (𝑋𝐶2) = {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} | ||
Theorem | clwwlknon2x 27810* | 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)} | ||
Theorem | s2elclwwlknon2 27811 | 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) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ {𝑋, 𝑌} ∈ 𝐸) → 〈“𝑋𝑌”〉 ∈ (𝑋𝐶2)) | ||
Theorem | clwwlknon2num 27812 | 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.) |
⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ (Vtx‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾) | ||
Theorem | clwwlknonwwlknonb 27813 | 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 27762. (Contributed by AV, 4-Mar-2022.) (Revised by AV, 27-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ++ 〈“𝑋”〉) ∈ (𝑋(𝑁 WWalksNOn 𝐺)𝑋))) | ||
Theorem | clwwlknonex2lem1 27814 | Lemma 1 for clwwlknonex2 27816: 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)})) | ||
Theorem | clwwlknonex2lem2 27815* | Lemma 2 for clwwlknonex2 27816: 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) ⇒ ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (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) | ||
Theorem | clwwlknonex2 27816 | 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) ⇒ ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ {𝑋, 𝑌} ∈ 𝐸 ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) → ((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ∈ (𝑁 ClWWalksN 𝐺)) | ||
Theorem | clwwlknonex2e 27817 | 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) ⇒ ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ {𝑋, 𝑌} ∈ 𝐸 ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) → ((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁)) | ||
Theorem | clwwlknondisj 27818* | 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 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁) | ||
Theorem | clwwlknun 27819* | 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) = ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁)) | ||
Theorem | clwwlkvbij 27820* | 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.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁)) | ||
Theorem | 0ewlk 27821 | 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)) | ||
Theorem | 1ewlk 27822 | 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)) | ||
Theorem | 0wlk 27823 | 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) ⇒ ⊢ (𝐺 ∈ 𝑈 → (∅(Walks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | ||
Theorem | is0wlk 27824 | 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〉} ∧ 𝑁 ∈ 𝑉) → ∅(Walks‘𝐺)𝑃) | ||
Theorem | 0wlkonlem1 27825 | Lemma 1 for 0wlkon 27827 and 0trlon 27831. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → (𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) | ||
Theorem | 0wlkonlem2 27826 | Lemma 2 for 0wlkon 27827 and 0trlon 27831. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → 𝑃 ∈ (𝑉 ↑pm (0...0))) | ||
Theorem | 0wlkon 27827 | 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) | ||
Theorem | 0wlkons1 27828 | A walk of length 0 from a vertex to itself. (Contributed by AV, 17-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑁 ∈ 𝑉 → ∅(𝑁(WalksOn‘𝐺)𝑁)〈“𝑁”〉) | ||
Theorem | 0trl 27829 | 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) ⇒ ⊢ (𝐺 ∈ 𝑈 → (∅(Trails‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | ||
Theorem | is0trl 27830 | 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〉} ∧ 𝑁 ∈ 𝑉) → ∅(Trails‘𝐺)𝑃) | ||
Theorem | 0trlon 27831 | 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) | ||
Theorem | 0pth 27832 | 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) ⇒ ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | ||
Theorem | 0spth 27833 | 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) ⇒ ⊢ (𝐺 ∈ 𝑊 → (∅(SPaths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | ||
Theorem | 0pthon 27834 | 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) | ||
Theorem | 0pthon1 27835 | 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) ⇒ ⊢ (𝑁 ∈ 𝑉 → ∅(𝑁(PathsOn‘𝐺)𝑁){〈0, 𝑁〉}) | ||
Theorem | 0pthonv 27836* | 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) ⇒ ⊢ (𝑁 ∈ 𝑉 → ∃𝑓∃𝑝 𝑓(𝑁(PathsOn‘𝐺)𝑁)𝑝) | ||
Theorem | 0clwlk 27837 | 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) ⇒ ⊢ (𝐺 ∈ 𝑋 → (∅(ClWalks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | ||
Theorem | 0clwlkv 27838 | 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) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹 = ∅ ∧ 𝑃:{0}⟶{𝑋}) → 𝐹(ClWalks‘𝐺)𝑃) | ||
Theorem | 0clwlk0 27839 | 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‘∅) = ∅ | ||
Theorem | 0crct 27840 | 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.) |
⊢ (𝐺 ∈ 𝑊 → (∅(Circuits‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶(Vtx‘𝐺))) | ||
Theorem | 0cycl 27841 | 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.) |
⊢ (𝐺 ∈ 𝑊 → (∅(Cycles‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶(Vtx‘𝐺))) | ||
Theorem | 1pthdlem1 27842 | Lemma 1 for 1pthd 27850. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 ⇒ ⊢ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) | ||
Theorem | 1pthdlem2 27843 | Lemma 2 for 1pthd 27850. (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)))) = ∅ | ||
Theorem | 1wlkdlem1 27844 | Lemma 1 for 1wlkd 27848. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉) | ||
Theorem | 1wlkdlem2 27845 | Lemma 2 for 1wlkd 27848. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐼‘𝐽)) | ||
Theorem | 1wlkdlem3 27846 | Lemma 3 for 1wlkd 27848. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) ⇒ ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼) | ||
Theorem | 1wlkdlem4 27847* | Lemma 4 for 1wlkd 27848. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) | ||
Theorem | 1wlkd 27848 | 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”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | 1trld 27849 | 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”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | ||
Theorem | 1pthd 27850 | 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”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) | ||
Theorem | 1pthond 27851 | 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”〉 & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃) | ||
Theorem | upgr1wlkdlem1 27852 | Lemma 1 for upgr1wlkd 27854. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) ⇒ ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋}) | ||
Theorem | upgr1wlkdlem2 27853 | Lemma 2 for upgr1wlkd 27854. (Contributed by AV, 22-Jan-2021.) |
⊢ 𝑃 = 〈“𝑋𝑌”〉 & ⊢ 𝐹 = 〈“𝐽”〉 & ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) ⇒ ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽)) | ||
Theorem | upgr1wlkd 27854 | 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”〉 & ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) & ⊢ (𝜑 → 𝐺 ∈ UPGraph) ⇒ ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | upgr1trld 27855 | 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”〉 & ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) & ⊢ (𝜑 → 𝐺 ∈ UPGraph) ⇒ ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | ||
Theorem | upgr1pthd 27856 | 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”〉 & ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) & ⊢ (𝜑 → 𝐺 ∈ UPGraph) ⇒ ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) | ||
Theorem | upgr1pthond 27857 | 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”〉 & ⊢ (𝜑 → 𝑋 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → 𝑌 ∈ (Vtx‘𝐺)) & ⊢ (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌}) & ⊢ (𝜑 → 𝐺 ∈ UPGraph) ⇒ ⊢ (𝜑 → 𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃) | ||
Theorem | lppthon 27858 | 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 ∧ (𝐼‘𝐽) = {𝐴}) → 〈“𝐽”〉(𝐴(PathsOn‘𝐺)𝐴)〈“𝐴𝐴”〉) | ||
Theorem | lp1cycl 27859 | 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 ∧ (𝐼‘𝐽) = {𝐴}) → 〈“𝐽”〉(Cycles‘𝐺)〈“𝐴𝐴”〉) | ||
Theorem | 1pthon2v 27860* | 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 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ ∃𝑒 ∈ 𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝) | ||
Theorem | 1pthon2ve 27861* | 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 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃𝑓∃𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝) | ||
Theorem | wlk2v2elem1 27862 | Lemma 1 for wlk2v2e 27864: 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 | ||
Theorem | wlk2v2elem2 27863* | Lemma 2 for wlk2v2e 27864: 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‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} | ||
Theorem | wlk2v2e 27864 | 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 𝑋 ≠ 𝑌. (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 | ||
Theorem | ntrl2v2e 27865 | 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 27864, but not a trail. Notice that 𝐺 is a simple graph (without loops) only if 𝑋 ≠ 𝑌. (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 | ||
Theorem | 3wlkdlem1 27866 | Lemma 1 for 3wlkd 27877. (Contributed by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 ⇒ ⊢ (♯‘𝑃) = ((♯‘𝐹) + 1) | ||
Theorem | 3wlkdlem2 27867 | Lemma 2 for 3wlkd 27877. (Contributed by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 ⇒ ⊢ (0..^(♯‘𝐹)) = {0, 1, 2} | ||
Theorem | 3wlkdlem3 27868 | Lemma 3 for 3wlkd 27877. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) ⇒ ⊢ (𝜑 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐷))) | ||
Theorem | 3wlkdlem4 27869* | Lemma 4 for 3wlkd 27877. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉) | ||
Theorem | 3wlkdlem5 27870* | Lemma 5 for 3wlkd 27877. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) | ||
Theorem | 3pthdlem1 27871* | Lemma 1 for 3pthd 27881. (Contributed by AV, 9-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗))) | ||
Theorem | 3wlkdlem6 27872 | Lemma 6 for 3wlkd 27877. (Contributed by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) ⇒ ⊢ (𝜑 → (𝐴 ∈ (𝐼‘𝐽) ∧ 𝐵 ∈ (𝐼‘𝐾) ∧ 𝐶 ∈ (𝐼‘𝐿))) | ||
Theorem | 3wlkdlem7 27873 | Lemma 7 for 3wlkd 27877. (Contributed by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) ⇒ ⊢ (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V)) | ||
Theorem | 3wlkdlem8 27874 | Lemma 8 for 3wlkd 27877. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) ⇒ ⊢ (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿)) | ||
Theorem | 3wlkdlem9 27875 | Lemma 9 for 3wlkd 27877. (Contributed by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) ⇒ ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ∧ {𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1)) ∧ {𝐶, 𝐷} ⊆ (𝐼‘(𝐹‘2)))) | ||
Theorem | 3wlkdlem10 27876* | Lemma 10 for 3wlkd 27877. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶𝐷”〉 & ⊢ 𝐹 = 〈“𝐽𝐾𝐿”〉 & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) | ||
Theorem | 3wlkd 27877 | 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 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | 3wlkond 27878 | 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 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐷)𝑃) | ||
Theorem | 3trld 27879 | 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 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) ⇒ ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | ||
Theorem | 3trlond 27880 | 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 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) ⇒ ⊢ (𝜑 → 𝐹(𝐴(TrailsOn‘𝐺)𝐷)𝑃) | ||
Theorem | 3pthd 27881 | 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 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) ⇒ ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) | ||
Theorem | 3pthond 27882 | 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 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) ⇒ ⊢ (𝜑 → 𝐹(𝐴(PathsOn‘𝐺)𝐷)𝑃) | ||
Theorem | 3spthd 27883 | 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 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) & ⊢ (𝜑 → 𝐴 ≠ 𝐷) ⇒ ⊢ (𝜑 → 𝐹(SPaths‘𝐺)𝑃) | ||
Theorem | 3spthond 27884 | 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 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) & ⊢ (𝜑 → 𝐴 ≠ 𝐷) ⇒ ⊢ (𝜑 → 𝐹(𝐴(SPathsOn‘𝐺)𝐷)𝑃) | ||
Theorem | 3cycld 27885 | 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 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) & ⊢ (𝜑 → 𝐴 = 𝐷) ⇒ ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃) | ||
Theorem | 3cyclpd 27886 | 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 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) & ⊢ (𝜑 → 𝐴 = 𝐷) ⇒ ⊢ (𝜑 → (𝐹(Cycles‘𝐺)𝑃 ∧ (♯‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴)) | ||
Theorem | upgr3v3e3cycl 27887* | 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) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐 ∧ 𝑐 ≠ 𝑎))) | ||
Theorem | uhgr3cyclexlem 27888 | Lemma for uhgr3cyclex 27889. (Contributed by AV, 12-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼‘𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼‘𝐾)))) → 𝐽 ≠ 𝐾) | ||
Theorem | uhgr3cyclex 27889* | 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 ∧ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)) | ||
Theorem | umgr3cyclex 27890* | 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 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴)) | ||
Theorem | umgr3v3e3cycl 27891* | 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 → (∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))) | ||
Theorem | upgr4cycl4dv4e 27892* | 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) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 ((({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ∧ ({𝑐, 𝑑} ∈ 𝐸 ∧ {𝑑, 𝑎} ∈ 𝐸)) ∧ ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑎 ≠ 𝑑) ∧ (𝑏 ≠ 𝑐 ∧ 𝑏 ≠ 𝑑 ∧ 𝑐 ≠ 𝑑)))) | ||
Syntax | cconngr 27893 | Extend class notation with connected graphs. |
class ConnGraph | ||
Definition | df-conngr 27894* | 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 27836 and dfconngr1 27895. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
⊢ ConnGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} | ||
Theorem | dfconngr1 27895* | 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]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ (𝑣 ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} | ||
Theorem | isconngr 27896* | 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) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ 𝑉 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) | ||
Theorem | isconngr1 27897* | 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) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝)) | ||
Theorem | cusconngr 27898 | 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) | ||
Theorem | 0conngr 27899 | A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
⊢ ∅ ∈ ConnGraph | ||
Theorem | 0vconngr 27900 | A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ ConnGraph) |
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