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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Definition | df-clwwlk 30001* | Define the set of all closed 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) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) as defined in df-clwlks 29791. Notice that the word does not contain the terminating vertex p(n) of the walk, because it is always equal to the first vertex of the closed walk. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) |
| ⊢ ClWWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))}) | ||
| Theorem | clwwlk 30002* | The set of closed walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (ClWWalks‘𝐺) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ 𝐸)} | ||
| Theorem | isclwwlk 30003* | Properties of a word to represent a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝑊 ∈ (ClWWalks‘𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)) | ||
| Theorem | clwwlkbp 30004 | Basic properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅)) | ||
| Theorem | clwwlkgt0 30005 | There is no empty closed walk (i.e. a closed walk without any edge) represented by a word of vertices. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.) |
| ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → 0 < (♯‘𝑊)) | ||
| Theorem | clwwlksswrd 30006 | Closed walks (represented by words) are words. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.) |
| ⊢ (ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺) | ||
| Theorem | clwwlk1loop 30007 | A closed walk of length 1 is a loop. See also clwlkl1loop 29803. (Contributed by AV, 24-Apr-2021.) |
| ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 1) → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺)) | ||
| Theorem | clwwlkccatlem 30008* | Lemma for clwwlkccat 30009: index 𝑗 is shifted up by (♯‘𝐴), and the case 𝑖 = ((♯‘𝐴) − 1) is covered by the "bridge" {(lastS‘𝐴), (𝐵‘0)} = {(lastS‘𝐴), (𝐴‘0)} ∈ (Edg‘𝐺). (Contributed by AV, 23-Apr-2022.) |
| ⊢ ((((𝐴 ∈ Word (Vtx‘𝐺) ∧ 𝐴 ≠ ∅) ∧ ∀𝑖 ∈ (0..^((♯‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝐴), (𝐴‘0)} ∈ (Edg‘𝐺)) ∧ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ≠ ∅) ∧ ∀𝑗 ∈ (0..^((♯‘𝐵) − 1)){(𝐵‘𝑗), (𝐵‘(𝑗 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝐵), (𝐵‘0)} ∈ (Edg‘𝐺)) ∧ (𝐴‘0) = (𝐵‘0)) → ∀𝑖 ∈ (0..^((♯‘(𝐴 ++ 𝐵)) − 1)){((𝐴 ++ 𝐵)‘𝑖), ((𝐴 ++ 𝐵)‘(𝑖 + 1))} ∈ (Edg‘𝐺)) | ||
| Theorem | clwwlkccat 30009 | The concatenation of two words representing closed walks anchored at the same vertex represents a closed walk. The resulting walk is a "double loop", starting at the common vertex, coming back to the common vertex by the first walk, following the second walk and finally coming back to the common vertex again. (Contributed by AV, 23-Apr-2022.) |
| ⊢ ((𝐴 ∈ (ClWWalks‘𝐺) ∧ 𝐵 ∈ (ClWWalks‘𝐺) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴 ++ 𝐵) ∈ (ClWWalks‘𝐺)) | ||
| Theorem | umgrclwwlkge2 30010 | A closed walk in a multigraph has a length of at least 2 (because it cannot have a loop). (Contributed by Alexander van der Vekens, 16-Sep-2018.) (Revised by AV, 24-Apr-2021.) |
| ⊢ (𝐺 ∈ UMGraph → (𝑃 ∈ (ClWWalks‘𝐺) → 2 ≤ (♯‘𝑃))) | ||
| Theorem | clwlkclwwlklem2a1 30011* | Lemma 1 for clwlkclwwlklem2a 30017. (Contributed by Alexander van der Vekens, 21-Jun-2018.) (Revised by AV, 11-Apr-2021.) |
| ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (((lastS‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((♯‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸)) | ||
| Theorem | clwlkclwwlklem2a2 30012* | Lemma 2 for clwlkclwwlklem2a 30017. (Contributed by Alexander van der Vekens, 21-Jun-2018.) |
| ⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) ⇒ ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (♯‘𝐹) = ((♯‘𝑃) − 1)) | ||
| Theorem | clwlkclwwlklem2a3 30013* | Lemma 3 for clwlkclwwlklem2a 30017. (Contributed by Alexander van der Vekens, 21-Jun-2018.) |
| ⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) ⇒ ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (𝑃‘(♯‘𝐹)) = (lastS‘𝑃)) | ||
| Theorem | clwlkclwwlklem2fv1 30014* | Lemma 4a for clwlkclwwlklem2a 30017. (Contributed by Alexander van der Vekens, 22-Jun-2018.) |
| ⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) ⇒ ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 2))) → (𝐹‘𝐼) = (◡𝐸‘{(𝑃‘𝐼), (𝑃‘(𝐼 + 1))})) | ||
| Theorem | clwlkclwwlklem2fv2 30015* | Lemma 4b for clwlkclwwlklem2a 30017. (Contributed by Alexander van der Vekens, 22-Jun-2018.) |
| ⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) ⇒ ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → (𝐹‘((♯‘𝑃) − 2)) = (◡𝐸‘{(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)})) | ||
| Theorem | clwlkclwwlklem2a4 30016* | Lemma 4 for clwlkclwwlklem2a 30017. (Contributed by Alexander van der Vekens, 21-Jun-2018.) (Revised by AV, 11-Apr-2021.) |
| ⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) ⇒ ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (((lastS‘𝑃) = (𝑃‘0) ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 1))) → ({(𝑃‘𝐼), (𝑃‘(𝐼 + 1))} ∈ ran 𝐸 → (𝐸‘(𝐹‘𝐼)) = {(𝑃‘𝐼), (𝑃‘(𝐼 + 1))}))) | ||
| Theorem | clwlkclwwlklem2a 30017* | Lemma for clwlkclwwlklem2 30019. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.) |
| ⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) ⇒ ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (((lastS‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((♯‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))) | ||
| Theorem | clwlkclwwlklem1 30018* | Lemma 1 for clwlkclwwlk 30021. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.) |
| ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (((lastS‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((♯‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∃𝑓((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(♯‘𝑓))))) | ||
| Theorem | clwlkclwwlklem2 30019* | Lemma 2 for clwlkclwwlk 30021. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.) |
| ⊢ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ 2 ≤ (♯‘𝑃)) ∧ (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) → ((lastS‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((♯‘𝐹) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝐹) − 1)), (𝑃‘0)} ∈ ran 𝐸)) | ||
| Theorem | clwlkclwwlklem3 30020* | Lemma 3 for clwlkclwwlk 30021. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.) |
| ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (∃𝑓((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(♯‘𝑓))) ↔ ((lastS‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((♯‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)))) | ||
| Theorem | clwlkclwwlk 30021* | A closed walk as word of length at least 2 corresponds to a closed walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 30-Oct-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)𝑃 ↔ ((lastS‘𝑃) = (𝑃‘0) ∧ (𝑃 prefix ((♯‘𝑃) − 1)) ∈ (ClWWalks‘𝐺)))) | ||
| Theorem | clwlkclwwlk2 30022* | A closed walk corresponds to a closed walk as word in a simple pseudograph. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 2-Nov-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑃 ++ 〈“(𝑃‘0)”〉) ↔ 𝑃 ∈ (ClWWalks‘𝐺))) | ||
| Theorem | clwlkclwwlkflem 30023* | Lemma for clwlkclwwlkf 30027. (Contributed by AV, 24-May-2022.) |
| ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐴 = (1st ‘𝑈) & ⊢ 𝐵 = (2nd ‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)) | ||
| Theorem | clwlkclwwlkf1lem2 30024* | Lemma 2 for clwlkclwwlkf1 30029. (Contributed by AV, 24-May-2022.) (Revised by AV, 30-Oct-2022.) |
| ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐴 = (1st ‘𝑈) & ⊢ 𝐵 = (2nd ‘𝑈) & ⊢ 𝐷 = (1st ‘𝑊) & ⊢ 𝐸 = (2nd ‘𝑊) ⇒ ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))) | ||
| Theorem | clwlkclwwlkf1lem3 30025* | Lemma 3 for clwlkclwwlkf1 30029. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 30-Oct-2022.) |
| ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐴 = (1st ‘𝑈) & ⊢ 𝐵 = (2nd ‘𝑈) & ⊢ 𝐷 = (1st ‘𝑊) & ⊢ 𝐸 = (2nd ‘𝑊) ⇒ ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ (0...(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)) | ||
| Theorem | clwlkclwwlkfolem 30026* | Lemma for clwlkclwwlkfo 30028. (Contributed by AV, 25-May-2022.) |
| ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ⇒ ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) → 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ 𝐶) | ||
| Theorem | clwlkclwwlkf 30027* | 𝐹 is a function from the nonempty closed walks into the closed walks as word in a simple pseudograph. (Contributed by AV, 23-May-2022.) (Revised by AV, 29-Oct-2022.) |
| ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) ⇒ ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺)) | ||
| Theorem | clwlkclwwlkfo 30028* | 𝐹 is a function from the nonempty closed walks onto the closed walks as words in a simple pseudograph. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by AV, 29-Oct-2022.) |
| ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) ⇒ ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–onto→(ClWWalks‘𝐺)) | ||
| Theorem | clwlkclwwlkf1 30029* | 𝐹 is a one-to-one function from the nonempty closed walks into the closed walks as words in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 29-Oct-2022.) |
| ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) ⇒ ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺)) | ||
| Theorem | clwlkclwwlkf1o 30030* | 𝐹 is a bijection between the nonempty closed walks and the closed walks as words in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 29-Oct-2022.) |
| ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) ⇒ ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1-onto→(ClWWalks‘𝐺)) | ||
| Theorem | clwlkclwwlken 30031* | The set of the nonempty closed walks and the set of closed walks as word are equinumerous in a simple pseudograph. (Contributed by AV, 25-May-2022.) (Proof shortened by AV, 4-Nov-2022.) |
| ⊢ (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ≈ (ClWWalks‘𝐺)) | ||
| Theorem | clwwisshclwwslemlem 30032* | Lemma for clwwisshclwwslem 30033. (Contributed by Alexander van der Vekens, 23-Mar-2018.) |
| ⊢ (((𝐿 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ∀𝑖 ∈ (0..^(𝐿 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝑅 ∧ {(𝑊‘(𝐿 − 1)), (𝑊‘0)} ∈ 𝑅) → {(𝑊‘((𝐴 + 𝐵) mod 𝐿)), (𝑊‘(((𝐴 + 1) + 𝐵) mod 𝐿))} ∈ 𝑅) | ||
| Theorem | clwwisshclwwslem 30033* | Lemma for clwwisshclwws 30034. (Contributed by AV, 24-Mar-2018.) (Revised by AV, 28-Apr-2021.) |
| ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(♯‘𝑊))) → ((∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) → ∀𝑗 ∈ (0..^((♯‘(𝑊 cyclShift 𝑁)) − 1)){((𝑊 cyclShift 𝑁)‘𝑗), ((𝑊 cyclShift 𝑁)‘(𝑗 + 1))} ∈ 𝐸)) | ||
| Theorem | clwwisshclwws 30034 | Cyclically shifting a closed walk as word results in a closed walk as word (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Mar-2018.) (Revised by AV, 28-Apr-2021.) |
| ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺)) | ||
| Theorem | clwwisshclwwsn 30035 | Cyclically shifting a closed walk as word results in a closed walk as word (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jun-2018.) (Revised by AV, 29-Apr-2021.) |
| ⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺)) | ||
| Theorem | erclwwlkrel 30036 | ∼ is a relation. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.) |
| ⊢ ∼ = {〈𝑢, 𝑤〉 ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))} ⇒ ⊢ Rel ∼ | ||
| Theorem | erclwwlkeq 30037* | Two classes are equivalent regarding ∼ if both are words and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.) |
| ⊢ ∼ = {〈𝑢, 𝑤〉 ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))} ⇒ ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 ↔ (𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)))) | ||
| Theorem | erclwwlkeqlen 30038* | If two classes are equivalent regarding ∼, then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by AV, 29-Apr-2021.) |
| ⊢ ∼ = {〈𝑢, 𝑤〉 ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))} ⇒ ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 → (♯‘𝑈) = (♯‘𝑊))) | ||
| Theorem | erclwwlkref 30039* | ∼ is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.) |
| ⊢ ∼ = {〈𝑢, 𝑤〉 ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))} ⇒ ⊢ (𝑥 ∈ (ClWWalks‘𝐺) ↔ 𝑥 ∼ 𝑥) | ||
| Theorem | erclwwlksym 30040* | ∼ is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by AV, 29-Apr-2021.) |
| ⊢ ∼ = {〈𝑢, 𝑤〉 ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))} ⇒ ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥) | ||
| Theorem | erclwwlktr 30041* | ∼ is a transitive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ ∼ = {〈𝑢, 𝑤〉 ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))} ⇒ ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧) | ||
| Theorem | erclwwlk 30042* | ∼ is an equivalence relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ ∼ = {〈𝑢, 𝑤〉 ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))} ⇒ ⊢ ∼ Er (ClWWalks‘𝐺) | ||
| Syntax | cclwwlkn 30043 | Extend class notation with closed walks (in an undirected graph) of a fixed length as word over the set of vertices. |
| class ClWWalksN | ||
| Definition | df-clwwlkn 30044* | Define the set of all closed walks of a fixed length 𝑛 as words over the set of vertices in a graph 𝑔. If 0 < 𝑛, such a word corresponds to the sequence p(0) p(1) ... p(n-1) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) as defined in df-clwlks 29791. For 𝑛 = 0, the set is empty, see clwwlkn0 30047. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2022.) |
| ⊢ ClWWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛}) | ||
| Theorem | clwwlkn 30045* | The set of closed walks of a fixed length 𝑁 as words over the set of vertices in a graph 𝐺. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2022.) |
| ⊢ (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} | ||
| Theorem | isclwwlkn 30046 | A word over the set of vertices representing a closed walk of a fixed length. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2022.) |
| ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁)) | ||
| Theorem | clwwlkn0 30047 | There is no closed walk of length 0 (i.e. a closed walk without any edge) represented by a word of vertices. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.) |
| ⊢ (0 ClWWalksN 𝐺) = ∅ | ||
| Theorem | clwwlkneq0 30048 | Sufficient conditions for ClWWalksN to be empty. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 24-Feb-2022.) |
| ⊢ ((𝐺 ∉ V ∨ 𝑁 ∉ ℕ) → (𝑁 ClWWalksN 𝐺) = ∅) | ||
| Theorem | clwwlkclwwlkn 30049 | A closed walk of a fixed length as word is a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) |
| ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ (ClWWalks‘𝐺)) | ||
| Theorem | clwwlksclwwlkn 30050 | The closed walks of a fixed length as words are closed walks (in an undirected graph) as words. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 12-Apr-2021.) |
| ⊢ (𝑁 ClWWalksN 𝐺) ⊆ (ClWWalks‘𝐺) | ||
| Theorem | clwwlknlen 30051 | The length of a word representing a closed walk of a fixed length is this fixed length. (Contributed by AV, 22-Mar-2022.) |
| ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (♯‘𝑊) = 𝑁) | ||
| Theorem | clwwlknnn 30052 | The length of a closed walk of a fixed length as word is a positive integer. (Contributed by AV, 22-Mar-2022.) |
| ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 ∈ ℕ) | ||
| Theorem | clwwlknwrd 30053 | A closed walk of a fixed length as word is a word over the vertices. (Contributed by AV, 30-Apr-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ Word 𝑉) | ||
| Theorem | clwwlknbp 30054 | Basic properties of a closed walk of a fixed length as word. (Contributed by AV, 30-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)) | ||
| Theorem | isclwwlknx 30055* | Characterization of a word representing a closed walk of a fixed length, definition of ClWWalks expanded. (Contributed by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (♯‘𝑊) = 𝑁))) | ||
| Theorem | clwwlknp 30056* | Properties of a set being a closed walk (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 23-Mar-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)) | ||
| Theorem | clwwlknwwlksn 30057 | A word representing a closed walk of length 𝑁 also represents a walk of length 𝑁 − 1. The walk is one edge shorter than the closed walk, because the last edge connecting the last with the first vertex is missing. For example, if 〈“𝑎𝑏𝑐”〉 ∈ (3 ClWWalksN 𝐺) represents a closed walk "abca" of length 3, then 〈“𝑎𝑏𝑐”〉 ∈ (2 WWalksN 𝐺) represents a walk "abc" (not closed if 𝑎 ≠ 𝑐) of length 2, and 〈“𝑎𝑏𝑐𝑎”〉 ∈ (3 WWalksN 𝐺) represents also a closed walk "abca" of length 3. (Contributed by AV, 24-Jan-2022.) (Revised by AV, 22-Mar-2022.) |
| ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ ((𝑁 − 1) WWalksN 𝐺)) | ||
| Theorem | clwwlknlbonbgr1 30058 | The last but one vertex in a closed walk is a neighbor of the first vertex of the closed walk. (Contributed by AV, 17-Feb-2022.) |
| ⊢ ((𝐺 ∈ USGraph ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑊‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx (𝑊‘0))) | ||
| Theorem | clwwlkinwwlk 30059 | If the initial vertex of a walk occurs another time in the walk, the walk starts with a closed walk. Since the walk is expressed as a word over vertices, the closed walk can be expressed as a subword of this word. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 23-Jan-2022.) (Revised by AV, 30-Oct-2022.) |
| ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) ∧ 𝑊 ∈ (𝑀 WWalksN 𝐺) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (𝑊 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺)) | ||
| Theorem | clwwlkn1 30060 | A closed walk of length 1 represented as word is a word consisting of 1 symbol representing a vertex connected to itself by (at least) one edge, that is, a loop. (Contributed by AV, 24-Apr-2021.) (Revised by AV, 11-Feb-2022.) |
| ⊢ (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺))) | ||
| Theorem | loopclwwlkn1b 30061 | The singleton word consisting of a vertex 𝑉 represents a closed walk of length 1 iff there is a loop at vertex 𝑉. (Contributed by AV, 11-Feb-2022.) |
| ⊢ (𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) ↔ 〈“𝑉”〉 ∈ (1 ClWWalksN 𝐺))) | ||
| Theorem | clwwlkn1loopb 30062* | A word represents a closed walk of length 1 iff this word is a singleton word consisting of a vertex with an attached loop. (Contributed by AV, 11-Feb-2022.) |
| ⊢ (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = 〈“𝑣”〉 ∧ {𝑣} ∈ (Edg‘𝐺))) | ||
| Theorem | clwwlkn2 30063 | A closed walk of length 2 represented as word is a word consisting of 2 symbols representing (not necessarily different) vertices connected by (at least) one edge. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Apr-2021.) |
| ⊢ (𝑊 ∈ (2 ClWWalksN 𝐺) ↔ ((♯‘𝑊) = 2 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0), (𝑊‘1)} ∈ (Edg‘𝐺))) | ||
| Theorem | clwwlknfi 30064 | If there is only a finite number of vertices, the number of closed walks of fixed length (as words) is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) (Proof shortened by JJ, 18-Nov-2022.) |
| ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) | ||
| Theorem | clwwlkel 30065* | Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 25-Apr-2021.) |
| ⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ⇒ ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))) → (𝑃 ++ 〈“(𝑃‘0)”〉) ∈ 𝐷) | ||
| Theorem | clwwlkf 30066* | Lemma 1 for clwwlkf1o 30070: F is a function. (Contributed by Alexander van der Vekens, 27-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.) |
| ⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁)) ⇒ ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺)) | ||
| Theorem | clwwlkfv 30067* | Lemma 2 for clwwlkf1o 30070: the value of function 𝐹. (Contributed by Alexander van der Vekens, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.) |
| ⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁)) ⇒ ⊢ (𝑊 ∈ 𝐷 → (𝐹‘𝑊) = (𝑊 prefix 𝑁)) | ||
| Theorem | clwwlkf1 30068* | Lemma 3 for clwwlkf1o 30070: 𝐹 is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.) |
| ⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁)) ⇒ ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–1-1→(𝑁 ClWWalksN 𝐺)) | ||
| Theorem | clwwlkfo 30069* | Lemma 4 for clwwlkf1o 30070: F is an onto function. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.) |
| ⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁)) ⇒ ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–onto→(𝑁 ClWWalksN 𝐺)) | ||
| Theorem | clwwlkf1o 30070* | F is a 1-1 onto function, that means that there is a bijection between the set of closed walks of a fixed length represented by walks (as words) and the set of closed walks (as words) of the fixed length. The difference between these two representations is that in the first case the starting 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, 1-Nov-2022.) |
| ⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁)) ⇒ ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–1-1-onto→(𝑁 ClWWalksN 𝐺)) | ||
| Theorem | clwwlken 30071* | The set of closed walks of a fixed length represented by walks (as words) and the set of closed walks (as words) of the fixed length are equinumerous. (Contributed by AV, 5-Jun-2022.) (Proof shortened by AV, 2-Nov-2022.) |
| ⊢ (𝑁 ∈ ℕ → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ≈ (𝑁 ClWWalksN 𝐺)) | ||
| Theorem | clwwlknwwlkncl 30072* | Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 22-Mar-2022.) |
| ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑊 ++ 〈“(𝑊‘0)”〉) ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}) | ||
| Theorem | clwwlkwwlksb 30073 | A nonempty word over vertices represents a closed walk iff the word concatenated with its first symbol represents a walk. (Contributed by AV, 4-Mar-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊 ∈ (ClWWalks‘𝐺) ↔ (𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (WWalks‘𝐺))) | ||
| Theorem | clwwlknwwlksnb 30074 | A word over vertices represents a closed walk of a fixed length 𝑁 greater than zero iff the word concatenated with its first symbol represents a walk of length 𝑁. This theorem would not hold for 𝑁 = 0 and 𝑊 = ∅, because (𝑊 ++ 〈“(𝑊‘0)”〉) = 〈“∅”〉 ∈ (0 WWalksN 𝐺) could be true, but not 𝑊 ∈ (0 ClWWalksN 𝐺) ↔ ∅ ∈ ∅. (Contributed by AV, 4-Mar-2022.) (Proof shortened by AV, 22-Mar-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ++ 〈“(𝑊‘0)”〉) ∈ (𝑁 WWalksN 𝐺))) | ||
| Theorem | clwwlkext2edg 30075 | If a word concatenated with a vertex represents a closed walk in (in a graph), there is an edge between this vertex and the last vertex of the word, and between this vertex and the first vertex of the word. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑍 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ (𝑊 ++ 〈“𝑍”〉) ∈ (𝑁 ClWWalksN 𝐺)) → ({(lastS‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸)) | ||
| Theorem | wwlksext2clwwlk 30076 | If a word represents a walk in (in a graph) and there are edges between the last vertex of the word and another vertex and between this other vertex and the first vertex of the word, then the concatenation of the word representing the walk with this other vertex represents a closed walk. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-Apr-2021.) (Revised by AV, 14-Mar-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑍 ∈ 𝑉) → (({(lastS‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸) → (𝑊 ++ 〈“𝑍”〉) ∈ ((𝑁 + 2) ClWWalksN 𝐺))) | ||
| Theorem | wwlksubclwwlk 30077 | Any prefix of a word representing a closed walk represents a walk. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Revised by AV, 28-Apr-2021.) (Revised by AV, 1-Nov-2022.) |
| ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑋 ∈ (𝑁 ClWWalksN 𝐺) → (𝑋 prefix 𝑀) ∈ ((𝑀 − 1) WWalksN 𝐺))) | ||
| Theorem | clwwnisshclwwsn 30078 | Cyclically shifting a closed walk as word of fixed length results in a closed walk as word of the same length (in an undirected graph). (Contributed by Alexander van der Vekens, 10-Jun-2018.) (Revised by AV, 29-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) |
| ⊢ ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺)) | ||
| Theorem | eleclclwwlknlem1 30079* | Lemma 1 for eleclclwwlkn 30095. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) ⇒ ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊)) → ((𝑋 = (𝑌 cyclShift 𝐾) ∧ ∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛))) | ||
| Theorem | eleclclwwlknlem2 30080* | Lemma 2 for eleclclwwlkn 30095. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) ⇒ ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))) | ||
| Theorem | clwwlknscsh 30081* | The set of cyclical shifts of a word representing a closed walk is the set of closed walks represented by cyclical shifts of a word. (Contributed by Alexander van der Vekens, 15-Jun-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)}) | ||
| Theorem | clwwlknccat 30082 | The concatenation of two words representing closed walks anchored at the same vertex represents a closed walk with a length which is the sum of the lengths of the two walks. The resulting walk is a "double loop", starting at the common vertex, coming back to the common vertex by the first walk, following the second walk and finally coming back to the common vertex again. (Contributed by AV, 24-Apr-2022.) |
| ⊢ ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐵 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴 ++ 𝐵) ∈ ((𝑀 + 𝑁) ClWWalksN 𝐺)) | ||
| Theorem | umgr2cwwk2dif 30083 | If a word represents a closed walk of length at least 2 in a multigraph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ (ℤ≥‘2) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑊‘1) ≠ (𝑊‘0)) | ||
| Theorem | umgr2cwwkdifex 30084* | If a word represents a closed walk of length at least 2 in a undirected simple graph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ (ℤ≥‘2) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) ≠ (𝑊‘0)) | ||
| Theorem | erclwwlknrel 30085 | ∼ is a relation. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ Rel ∼ | ||
| Theorem | erclwwlkneq 30086* | Two classes are equivalent regarding ∼ if both are words of the same fixed length and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)))) | ||
| Theorem | erclwwlkneqlen 30087* | If two classes are equivalent regarding ∼, then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 → (♯‘𝑇) = (♯‘𝑈))) | ||
| Theorem | erclwwlknref 30088* | ∼ is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 26-Mar-2018.) (Revised by AV, 30-Apr-2021.) (Proof shortened by AV, 23-Mar-2022.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∼ 𝑥) | ||
| Theorem | erclwwlknsym 30089* | ∼ is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥) | ||
| Theorem | erclwwlkntr 30090* | ∼ is a transitive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧) | ||
| Theorem | erclwwlkn 30091* | ∼ is an equivalence relation over the set of closed walks (defined as words) with a fixed length. (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ ∼ Er 𝑊 | ||
| Theorem | qerclwwlknfi 30092* | The quotient set of the set of closed walks (defined as words) with a fixed length according to the equivalence relation ∼ is finite. (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑊 / ∼ ) ∈ Fin) | ||
| Theorem | hashclwwlkn0 30093* | The number of closed walks (defined as words) with a fixed length is the sum of the sizes of all equivalence classes according to ∼. (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ ((Vtx‘𝐺) ∈ Fin → (♯‘𝑊) = Σ𝑥 ∈ (𝑊 / ∼ )(♯‘𝑥)) | ||
| Theorem | eclclwwlkn1 30094* | An equivalence class according to ∼. (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)})) | ||
| Theorem | eleclclwwlkn 30095* | A member of an equivalence class according to ∼. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 1-May-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ ((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑋 ∈ 𝐵) → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))) | ||
| Theorem | hashecclwwlkn1 30096* | The size of every equivalence class of the equivalence relation over the set of closed walks (defined as words) with a fixed length which is a prime number is 1 or equals this length. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 1-May-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ ((𝑁 ∈ ℙ ∧ 𝑈 ∈ (𝑊 / ∼ )) → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁)) | ||
| Theorem | umgrhashecclwwlk 30097* | The size of every equivalence class of the equivalence relation over the set of closed walks (defined as words) with a fixed length which is a prime number equals this length (in an undirected simple graph). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 1-May-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 ∈ (𝑊 / ∼ ) → (♯‘𝑈) = 𝑁)) | ||
| Theorem | fusgrhashclwwlkn 30098* | The size of the set of closed walks (defined as words) with a fixed length which is a prime number is the product of the number of equivalence classes for ∼ over the set of closed walks and the fixed length. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 1-May-2021.) |
| ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) & ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (♯‘𝑊) = ((♯‘(𝑊 / ∼ )) · 𝑁)) | ||
| Theorem | clwwlkndivn 30099 | The size of the set of closed walks (defined as words) of length 𝑁 is divisible by 𝑁 if 𝑁 is a prime number. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 2-May-2021.) |
| ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∥ (♯‘(𝑁 ClWWalksN 𝐺))) | ||
| Theorem | clwlknf1oclwwlknlem1 30100 | Lemma 1 for clwlknf1oclwwlkn 30103. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.) |
| ⊢ ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶))) | ||
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