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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | grlicrel 47901 | The "is locally isomorphic to" relation for graphs is a relation. (Contributed by AV, 9-Jun-2025.) |
⊢ Rel ≃𝑙𝑔𝑟 | ||
Theorem | grlicrcl 47902 | Reverse closure of the "is locally isomorphic to" relation for graphs. (Contributed by AV, 9-Jun-2025.) |
⊢ (𝐺 ≃𝑙𝑔𝑟 𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V)) | ||
Theorem | dfgrlic2 47903* | Alternate, explicit definition of the "is locally isomorphic to" relation for two graphs. (Contributed by AV, 9-Jun-2025.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑊 = (Vtx‘𝐻) ⇒ ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣)))))) | ||
Theorem | grilcbri 47904* | Implications of two graphs being locally isomorphic. (Contributed by AV, 9-Jun-2025.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑊 = (Vtx‘𝐻) ⇒ ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣))))) | ||
Theorem | dfgrlic3 47905* | Alternate, explicit definition of the "is locally isomorphic to" relation for two graphs. (Contributed by AV, 9-Jun-2025.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑊 = (Vtx‘𝐻) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐽 = (iEdg‘𝐻) & ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣) & ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑣)) & ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁} & ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀} ⇒ ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))) | ||
Theorem | grilcbri2 47906* | Implications of two graphs being locally isomorphic. (Contributed by AV, 9-Jun-2025.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑊 = (Vtx‘𝐻) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐽 = (iEdg‘𝐻) & ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋) & ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑋)) & ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁} & ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀} ⇒ ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ (𝑋 ∈ 𝑉 → ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))) | ||
Theorem | grlicref 47907 | Graph local isomorphism is reflexive for hypergraphs. (Contributed by AV, 9-Jun-2025.) |
⊢ (𝐺 ∈ UHGraph → 𝐺 ≃𝑙𝑔𝑟 𝐺) | ||
Theorem | grlicsym 47908 | Graph local isomorphism is symmetric for hypergraphs. (Contributed by AV, 9-Jun-2025.) |
⊢ (𝐺 ∈ UHGraph → (𝐺 ≃𝑙𝑔𝑟 𝑆 → 𝑆 ≃𝑙𝑔𝑟 𝐺)) | ||
Theorem | grlicsymb 47909 | Graph local isomorphism is symmetric in both directions for hypergraphs. (Contributed by AV, 9-Jun-2025.) |
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 ≃𝑙𝑔𝑟 𝐵 ↔ 𝐵 ≃𝑙𝑔𝑟 𝐴)) | ||
Theorem | grlictr 47910 | Graph local isomorphism is transitive. (Contributed by AV, 10-Jun-2025.) |
⊢ ((𝑅 ≃𝑙𝑔𝑟 𝑆 ∧ 𝑆 ≃𝑙𝑔𝑟 𝑇) → 𝑅 ≃𝑙𝑔𝑟 𝑇) | ||
Theorem | grlicer 47911 | Local isomorphism is an equivalence relation on hypergraphs. (Contributed by AV, 11-Jun-2025.) |
⊢ ( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph | ||
Theorem | grlicen 47912 | Locally isomorphic graphs have equinumerous sets of vertices. (Contributed by AV, 11-Jun-2025.) |
⊢ 𝐵 = (Vtx‘𝑅) & ⊢ 𝐶 = (Vtx‘𝑆) ⇒ ⊢ (𝑅 ≃𝑙𝑔𝑟 𝑆 → 𝐵 ≈ 𝐶) | ||
Theorem | gricgrlic 47913 | Isomorphic hypergraphs are locally isomorphic. (Contributed by AV, 12-Jun-2025.) (Proof shortened by AV, 11-Jul-2025.) |
⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐺 ≃𝑔𝑟 𝐻 → 𝐺 ≃𝑙𝑔𝑟 𝐻)) | ||
Theorem | clnbgr3stgrgrlic 47914* | If all (closed) neighborhoods of the vertices in two simple graphs with the same order induce a subgraph which is isomorphic to an 𝑁-star, then the two graphs are locally isomorphic. (Contributed by AV, 29-Sep-2025.) |
⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑊 = (Vtx‘𝐻) ⇒ ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐺 ≃𝑙𝑔𝑟 𝐻) | ||
Theorem | usgrexmpl1lem 47915* | Lemma for usgrexmpl1 47916. (Contributed by AV, 2-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”〉 ⇒ ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} | ||
Theorem | usgrexmpl1 47916 | 𝐺 is a simple graph of six vertices 0, 1, 2, 3, 4, 5, with edges {0, 1}, {1, 2}, {0, 2}, {0, 3}, {3, 4}, {3, 5}, {4, 5}. (Contributed by AV, 3-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ 𝐺 ∈ USGraph | ||
Theorem | usgrexmpl1vtx 47917 | The vertices 0, 1, 2, 3, 4, 5 of the graph 𝐺 = 〈𝑉, 𝐸〉. (Contributed by AV, 3-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (Vtx‘𝐺) = ({0, 1, 2} ∪ {3, 4, 5}) | ||
Theorem | usgrexmpl1edg 47918 | The edges {0, 1}, {1, 2}, {0, 2}, {0, 3}, {3, 4}, {3, 5}, {4, 5} of the graph 𝐺 = 〈𝑉, 𝐸〉. (Contributed by AV, 3-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (Edg‘𝐺) = ({{0, 3}} ∪ ({{0, 1}, {0, 2}, {1, 2}} ∪ {{3, 4}, {3, 5}, {4, 5}})) | ||
Theorem | usgrexmpl1tri 47919 | 𝐺 contains a triangle 0, 1, 2, with corresponding edges {0, 1}, {1, 2}, {0, 2}. (Contributed by AV, 3-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ {0, 1, 2} ∈ (GrTriangles‘𝐺) | ||
Theorem | usgrexmpl2lem 47920* | Lemma for usgrexmpl2 47921. (Contributed by AV, 3-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 ⇒ ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} | ||
Theorem | usgrexmpl2 47921 | 𝐺 is a simple graph of six vertices 0, 1, 2, 3, 4, 5, with edges {0, 1}, {1, 2}, {2, 3}, {0, 3}, {3, 4}, {4, 5}, {0, 5}. (Contributed by AV, 3-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ 𝐺 ∈ USGraph | ||
Theorem | usgrexmpl2vtx 47922 | The vertices 0, 1, 2, 3, 4, 5 of the graph 𝐺 = 〈𝑉, 𝐸〉. (Contributed by AV, 3-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (Vtx‘𝐺) = ({0, 1, 2} ∪ {3, 4, 5}) | ||
Theorem | usgrexmpl2edg 47923 | The edges {0, 1}, {1, 2}, {2, 3}, {0, 3}, {3, 4}, {4, 5}, {0, 5} of the graph 𝐺 = 〈𝑉, 𝐸〉. (Contributed by AV, 3-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (Edg‘𝐺) = ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) | ||
Theorem | usgrexmpl2nblem 47924* | Lemma for usgrexmpl2nb0 47925 etc. (Contributed by AV, 9-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (𝐾 ∈ ({0, 1, 2} ∪ {3, 4, 5}) → (𝐺 NeighbVtx 𝐾) = {𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∣ {𝐾, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))}) | ||
Theorem | usgrexmpl2nb0 47925 | The neighborhood of the first vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (𝐺 NeighbVtx 0) = {1, 3, 5} | ||
Theorem | usgrexmpl2nb1 47926 | The neighborhood of the second vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (𝐺 NeighbVtx 1) = {0, 2} | ||
Theorem | usgrexmpl2nb2 47927 | The neighborhood of the third vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (𝐺 NeighbVtx 2) = {1, 3} | ||
Theorem | usgrexmpl2nb3 47928 | The neighborhood of the forth vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (𝐺 NeighbVtx 3) = {0, 2, 4} | ||
Theorem | usgrexmpl2nb4 47929 | The neighborhood of the fifth vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (𝐺 NeighbVtx 4) = {3, 5} | ||
Theorem | usgrexmpl2nb5 47930 | The neighborhood of the sixth vertex of graph 𝐺. (Contributed by AV, 10-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (𝐺 NeighbVtx 5) = {0, 4} | ||
Theorem | usgrexmpl2trifr 47931* | 𝐺 is triangle-free. (Contributed by AV, 10-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ ¬ ∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) | ||
Theorem | usgrexmpl12ngric 47932 | The graphs 𝐻 and 𝐺 are not isomorphic (𝐻 contains a triangle, see usgrexmpl1tri 47919, whereas 𝐺 does not, see usgrexmpl2trifr 47931. (Contributed by AV, 10-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 & ⊢ 𝐾 = 〈“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”〉 & ⊢ 𝐻 = 〈𝑉, 𝐾〉 ⇒ ⊢ ¬ 𝐺 ≃𝑔𝑟 𝐻 | ||
Theorem | usgrexmpl12ngrlic 47933 | The graphs 𝐻 and 𝐺 are not locally isomorphic (𝐻 contains a triangle, see usgrexmpl1tri 47919, whereas 𝐺 does not, see usgrexmpl2trifr 47931. (Contributed by AV, 24-Aug-2025.) |
⊢ 𝑉 = (0...5) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 & ⊢ 𝐾 = 〈“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”〉 & ⊢ 𝐻 = 〈𝑉, 𝐾〉 ⇒ ⊢ ¬ 𝐺 ≃𝑙𝑔𝑟 𝐻 | ||
According to Wikipedia "Generalized Petersen graph", 26-Aug-2025, https://en.wikipedia.org/wiki/Generalized_Petersen_graph: "In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. ... Among the generalized Petersen graphs are the n-prism, ...". The vertices of the regular polygon are called "vertices of the first kind", the vertices of the star polygon "vertices of the second kind". Since regular polygons are also considered as star polygons (with density 1), many theorems for "vertices of the second kind" (with labels containing the fragment "vtx0") can be specialized for "vertices of the first kind" (with labels containing the fragment "vtx1"). | ||
Syntax | cgpg 47934 | Extend class notation with generalized Petersen graphs. |
class gPetersenGr | ||
Definition | df-gpg 47935* |
Definition of generalized Petersen graphs according to Wikipedia
"Generalized Petersen graph", 26-Aug-2025,
https://en.wikipedia.org/wiki/Generalized_Petersen_graph:
"In
Watkins' notation, 𝐺(𝑛, 𝑘) is a graph with vertex set {
u0,
u1, ... , un-1, v0, v1, ... , vn-1 } and
edge set { ui ui+1 , ui
vi , vi vi+k | 0 ≤ 𝑖 ≤
(𝑛 − 1) }
where subscripts are to be
read modulo n and where 𝑘 < (𝑛 / 2). Some authors use the
notation GPG(n,k)."
Instead of 𝑛 ∈ ℕ, we could restrict the first argument to 𝑛 ∈ (ℤ≥‘3) (i.e., 3 ≤ 𝑛), because for 𝑛 ≤ 2, the definition is not meaningful (since then (⌈‘(𝑛 / 2)) ≤ 1 and therefore (1..^(⌈‘(𝑛 / 2))) = ∅, so that there would be no fitting second argument). (Contributed by AV, 26-Aug-2025.) |
⊢ gPetersenGr = (𝑛 ∈ ℕ, 𝑘 ∈ (1..^(⌈‘(𝑛 / 2))) ↦ {〈(Base‘ndx), ({0, 1} × (0..^𝑛))〉, 〈(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × (0..^𝑛)) ∣ ∃𝑥 ∈ (0..^𝑛)(𝑒 = {〈0, 𝑥〉, 〈0, ((𝑥 + 1) mod 𝑛)〉} ∨ 𝑒 = {〈0, 𝑥〉, 〈1, 𝑥〉} ∨ 𝑒 = {〈1, 𝑥〉, 〈1, ((𝑥 + 𝑘) mod 𝑛)〉})})〉}) | ||
Theorem | gpgov 47936* | The generalized Petersen graph GPG(N,K). (Contributed by AV, 26-Aug-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐼 = (0..^𝑁) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (𝑁 gPetersenGr 𝐾) = {〈(Base‘ndx), ({0, 1} × 𝐼)〉, 〈(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {〈0, 𝑥〉, 〈0, ((𝑥 + 1) mod 𝑁)〉} ∨ 𝑒 = {〈0, 𝑥〉, 〈1, 𝑥〉} ∨ 𝑒 = {〈1, 𝑥〉, 〈1, ((𝑥 + 𝐾) mod 𝑁)〉})})〉}) | ||
Theorem | gpgvtx 47937 | The vertices of the generalized Petersen graph GPG(N,K). (Contributed by AV, 26-Aug-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐼 = (0..^𝑁) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (Vtx‘(𝑁 gPetersenGr 𝐾)) = ({0, 1} × 𝐼)) | ||
Theorem | gpgiedg 47938* | The indexed edges of the generalized Petersen graph GPG(N,K). (Contributed by AV, 26-Aug-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐼 = (0..^𝑁) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (iEdg‘(𝑁 gPetersenGr 𝐾)) = ( I ↾ {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {〈0, 𝑥〉, 〈0, ((𝑥 + 1) mod 𝑁)〉} ∨ 𝑒 = {〈0, 𝑥〉, 〈1, 𝑥〉} ∨ 𝑒 = {〈1, 𝑥〉, 〈1, ((𝑥 + 𝐾) mod 𝑁)〉})})) | ||
Theorem | gpgedg 47939* | The edges of the generalized Petersen graph GPG(N,K). (Contributed by AV, 26-Aug-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐼 = (0..^𝑁) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (Edg‘(𝑁 gPetersenGr 𝐾)) = {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {〈0, 𝑥〉, 〈0, ((𝑥 + 1) mod 𝑁)〉} ∨ 𝑒 = {〈0, 𝑥〉, 〈1, 𝑥〉} ∨ 𝑒 = {〈1, 𝑥〉, 〈1, ((𝑥 + 𝐾) mod 𝑁)〉})}) | ||
Theorem | gpgvtxel 47940* | A vertex in a generalized Petersen graph 𝐺. (Contributed by AV, 29-Aug-2025.) |
⊢ 𝐼 = (0..^𝑁) & ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ 𝑉 ↔ ∃𝑥 ∈ {0, 1}∃𝑦 ∈ 𝐼 𝑋 = 〈𝑥, 𝑦〉)) | ||
Theorem | gpgvtxel2 47941 | The second component of a vertex in a generalized Petersen graph 𝐺. (Contributed by AV, 30-Aug-2025.) |
⊢ 𝐼 = (0..^𝑁) & ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (2nd ‘𝑋) ∈ 𝐼) | ||
Theorem | gpgedgel 47942* | An edge in a generalized Petersen graph 𝐺. (Contributed by AV, 29-Aug-2025.) |
⊢ 𝐼 = (0..^𝑁) & ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (𝑌 ∈ 𝐸 ↔ ∃𝑥 ∈ 𝐼 (𝑌 = {〈0, 𝑥〉, 〈0, ((𝑥 + 1) mod 𝑁)〉} ∨ 𝑌 = {〈0, 𝑥〉, 〈1, 𝑥〉} ∨ 𝑌 = {〈1, 𝑥〉, 〈1, ((𝑥 + 𝐾) mod 𝑁)〉}))) | ||
Theorem | gpgvtx0 47943 | The vertices of the first kind in a generalized Petersen graph 𝐺. (Contributed by AV, 30-Aug-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (〈0, (((2nd ‘𝑋) + 1) mod 𝑁)〉 ∈ 𝑉 ∧ 〈0, (2nd ‘𝑋)〉 ∈ 𝑉 ∧ 〈0, (((2nd ‘𝑋) − 1) mod 𝑁)〉 ∈ 𝑉)) | ||
Theorem | gpgvtx1 47944 | The vertices of the second kind in a generalized Petersen graph 𝐺. (Contributed by AV, 28-Aug-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑉) → (〈1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)〉 ∈ 𝑉 ∧ 〈1, (2nd ‘𝑋)〉 ∈ 𝑉 ∧ 〈1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)〉 ∈ 𝑉)) | ||
Theorem | gpgusgralem 47945* | Lemma for gpgusgra 47946. (Contributed by AV, 27-Aug-2025.) (Proof shortened by AV, 6-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐼 = (0..^𝑁) ⇒ ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → {𝑒 ∈ 𝒫 ({0, 1} × 𝐼) ∣ ∃𝑥 ∈ 𝐼 (𝑒 = {〈0, 𝑥〉, 〈0, ((𝑥 + 1) mod 𝑁)〉} ∨ 𝑒 = {〈0, 𝑥〉, 〈1, 𝑥〉} ∨ 𝑒 = {〈1, 𝑥〉, 〈1, ((𝑥 + 𝐾) mod 𝑁)〉})} ⊆ {𝑝 ∈ 𝒫 ({0, 1} × 𝐼) ∣ (♯‘𝑝) = 2}) | ||
Theorem | gpgusgra 47946 | The generalized Petersen graph GPG(N,K) is a simple graph. (Contributed by AV, 27-Aug-2025.) |
⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (𝑁 gPetersenGr 𝐾) ∈ USGraph) | ||
Theorem | gpgorder 47947 | The order of the generalized Petersen graph GPG(N,K). (Contributed by AV, 29-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (♯‘(Vtx‘(𝑁 gPetersenGr 𝐾))) = (2 · 𝑁)) | ||
Theorem | gpg5order 47948 | The order of a generalized Petersen graph G(5,K), which is either the Petersen graph G(5,2) or the 5-prism G(5,1), is 10. (Contributed by AV, 26-Aug-2025.) |
⊢ (𝐾 ∈ (1...2) → (♯‘(Vtx‘(5 gPetersenGr 𝐾))) = ;10) | ||
Theorem | 2ltceilhalf 47949 | The ceiling of half of an integer greater than 2 is greater than or equal to 2. (Contributed by AV, 4-Sep-2025.) |
⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ≤ (⌈‘(𝑁 / 2))) | ||
Theorem | ceilhalfelfzo1 47950 | A positive integer less than (the ceiling of) half of another integer is in the half-open range of positive integers up to the other integer. (Contributed by AV, 7-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) ⇒ ⊢ (𝑁 ∈ ℕ → (𝐾 ∈ 𝐽 → 𝐾 ∈ (1..^𝑁))) | ||
Theorem | gpgedgvtx1lem 47951 | Lemma for gpgedgvtx1 47954. (Contributed by AV, 1-Sep-2025.) (Proof shortened by AV, 8-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐼 = (0..^𝑁) ⇒ ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑋 ∈ 𝐽) → 𝑋 ∈ 𝐼) | ||
Theorem | 2tceilhalfelfzo1 47952 | Two times a positive integer less than (the ceiling of) half of another integer is less than the other integer. This theorem would hold even for integers less than 3, but then a corresponding 𝐾 would not exist. (Contributed by AV, 9-Sep-2025.) |
⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2)))) → (2 · 𝐾) < 𝑁) | ||
Theorem | gpgedgvtx0 47953 | The edges starting at a vertex of the first kind in a generalized Petersen graph 𝐺. (Contributed by AV, 29-Aug-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ({𝑋, 〈0, (((2nd ‘𝑋) + 1) mod 𝑁)〉} ∈ 𝐸 ∧ {𝑋, 〈1, (2nd ‘𝑋)〉} ∈ 𝐸 ∧ {𝑋, 〈0, (((2nd ‘𝑋) − 1) mod 𝑁)〉} ∈ 𝐸)) | ||
Theorem | gpgedgvtx1 47954 | The edges starting at a vertex of the second kind in a generalized Petersen graph 𝐺. (Contributed by AV, 2-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → ({𝑋, 〈1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)〉} ∈ 𝐸 ∧ {𝑋, 〈0, (2nd ‘𝑋)〉} ∈ 𝐸 ∧ {𝑋, 〈1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)〉} ∈ 𝐸)) | ||
Theorem | gpgvtxedg0 47955 | The edges starting at a vertex 𝑋 of the first kind in a generalized Petersen graph 𝐺. (Contributed by AV, 30-Aug-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 0 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑌 = 〈0, (((2nd ‘𝑋) + 1) mod 𝑁)〉 ∨ 𝑌 = 〈1, (2nd ‘𝑋)〉 ∨ 𝑌 = 〈0, (((2nd ‘𝑋) − 1) mod 𝑁)〉)) | ||
Theorem | gpgvtxedg1 47956 | The edges starting at a vertex 𝑋 of the second kind in a generalized Petersen graph 𝐺. (Contributed by AV, 2-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (1st ‘𝑋) = 1 ∧ {𝑋, 𝑌} ∈ 𝐸) → (𝑌 = 〈1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)〉 ∨ 𝑌 = 〈0, (2nd ‘𝑋)〉 ∨ 𝑌 = 〈1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)〉)) | ||
Theorem | gpg3nbgrvtxlem 47957 | Lemma for gpg3nbgrvtx0ALT 47967 and gpg3nbgrvtx1 47968. For this theorem, it is essential that 2 < 𝑁 and 𝐾 < (𝑁 / 2)! (Contributed by AV, 3-Sep-2025.) (Proof shortened by AV, 9-Sep-2025.) |
⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))) ∧ 𝐴 ∈ (0..^𝑁)) → ((𝐴 + 𝐾) mod 𝑁) ≠ ((𝐴 − 𝐾) mod 𝑁)) | ||
Theorem | gpg5nbgrvtx03starlem1 47958 | Lemma 1 for gpg5nbgrvtx03star 47970. (Contributed by AV, 5-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊) → {〈0, ((𝑋 + 1) mod 𝑁)〉, 〈1, 𝑋〉} ∉ 𝐸) | ||
Theorem | gpg5nbgrvtx03starlem2 47959 | Lemma 2 for gpg5nbgrvtx03star 47970. (Contributed by AV, 6-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ ℤ) → {〈0, ((𝑋 + 1) mod 𝑁)〉, 〈0, ((𝑋 − 1) mod 𝑁)〉} ∉ 𝐸) | ||
Theorem | gpg5nbgrvtx03starlem3 47960 | Lemma 3 for gpg5nbgrvtx03star 47970. (Contributed by AV, 5-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑊) → {〈1, 𝑋〉, 〈0, ((𝑋 − 1) mod 𝑁)〉} ∉ 𝐸) | ||
Theorem | gpg5nbgrvtx13starlem1 47961 | Lemma 1 for gpg5nbgr3star 47971. (Contributed by AV, 7-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑊) → {〈1, ((𝑋 + 𝐾) mod 𝑁)〉, 〈0, 𝑋〉} ∉ 𝐸) | ||
Theorem | gpg5nbgrvtx13starlem2 47962 | Lemma 2 for gpg5nbgr3star 47971. (Contributed by AV, 8-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ ℤ) → {〈1, ((𝑋 + 𝐾) mod 𝑁)〉, 〈1, ((𝑋 − 𝐾) mod 𝑁)〉} ∉ 𝐸) | ||
Theorem | gpg5nbgrvtx13starlem3 47963 | Lemma 3 for gpg5nbgr3star 47971. (Contributed by AV, 8-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝑁 = 5 ∧ 𝐾 ∈ 𝐽) ∧ 𝑋 ∈ 𝑊) → {〈0, 𝑋〉, 〈1, ((𝑋 − 𝐾) mod 𝑁)〉} ∉ 𝐸) | ||
Theorem | gpgnbgrvtx0 47964 | The (open) neighborhood of a vertex of the first kind in a generalized Petersen graph 𝐺. (Contributed by AV, 28-Aug-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → 𝑈 = {〈0, (((2nd ‘𝑋) + 1) mod 𝑁)〉, 〈1, (2nd ‘𝑋)〉, 〈0, (((2nd ‘𝑋) − 1) mod 𝑁)〉}) | ||
Theorem | gpgnbgrvtx1 47965 | The (open) neighborhood of a vertex of the second kind in a generalized Petersen graph 𝐺. (Contributed by AV, 2-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → 𝑈 = {〈1, (((2nd ‘𝑋) + 𝐾) mod 𝑁)〉, 〈0, (2nd ‘𝑋)〉, 〈1, (((2nd ‘𝑋) − 𝐾) mod 𝑁)〉}) | ||
Theorem | gpg3nbgrvtx0 47966 | In a generalized Petersen graph 𝐺, every vertex of the first kind has exactly three (different) neighbors. (Contributed by AV, 30-Aug-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3) | ||
Theorem | gpg3nbgrvtx0ALT 47967 | In a generalized Petersen graph 𝐺, every vertex of the first kind has exactly three (different) neighbors. (Contributed by AV, 30-Aug-2025.) The proof of gpg3nbgrvtx0 47966 can be shortened using lemma gpg3nbgrvtxlem 47957, but then theorem 2ltceilhalf 47949 is required which is based on an "example" ex-ceil 30476. If these theorems were moved to main, the "example" should also be moved up to become a full-fledged theorem. (Proof shortened by AV, 4-Sep-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → (♯‘𝑈) = 3) | ||
Theorem | gpg3nbgrvtx1 47968 | In a generalized Petersen graph 𝐺, every vertex of the second kind has exactly three (different) neighbors. (Contributed by AV, 3-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 1)) → (♯‘𝑈) = 3) | ||
Theorem | gpgcubic 47969 | Every generalized Petersen graph is a cubic graph, i.e., it is a 3-regular graph, i.e., every vertex has degree 3 (see gpgvtxdg3 47972), i.e., every vertex has exactly three (different) neighbors. (Contributed by AV, 3-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋) ⇒ ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → (♯‘𝑈) = 3) | ||
Theorem | gpg5nbgrvtx03star 47970* | In a generalized Petersen graph G(N,K) of order greater than 8 (3 < 𝑁), every vertex of the first kind has exactly three (different) neighbors, and none of these neighbors are connected by an edge (i.e., the (closed) neighborhood of every vertex of the first kind induces a subgraph which is isomorphic to a 3-star). (Contributed by AV, 31-Aug-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝑁 ∈ (ℤ≥‘4) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝑉 ∧ (1st ‘𝑋) = 0)) → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) | ||
Theorem | gpg5nbgr3star 47971* | In a generalized Petersen graph G(N,K) of order 10 (𝑁 = 5), these are the Petersen graph G(5,2) and the 5-prism G(5,1), every vertex has exactly three (different) neighbors, and none of these neighbors are connected by an edge (i.e., the (closed) neighborhood of every vertex induces a subgraph which is isomorphic to a 3-star). This does not hold for every generalized Petersen graph: for example, in the 3-prism G(3,1) (see gpg31grim3prism TODO) and the Dürer graph G(6,2) there are vertices which have neighborhoods containing triangles. In general, all generalized Petersen graphs G(N,K) with 𝑁 = 3 · 𝐾 contain triangles (see gpg3kgrtriex TODO). (Contributed by AV, 8-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑈 = (𝐺 NeighbVtx 𝑋) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝑁 = 5 ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((♯‘𝑈) = 3 ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∉ 𝐸)) | ||
Theorem | gpgvtxdg3 47972 | Every vertex in a generalized Petersen graph has degree 3. (Contributed by AV, 4-Sep-2025.) |
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2))) & ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽 ∧ 𝑋 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑋) = 3) | ||
Theorem | gpg5gricstgr3 47973 | Each closed neighborhood in a generalized Petersen graph G(N,K) of order 10 (𝑁 = 5), which is either the Petersen graph G(5,2) or the 5-prism G(5,1), is isomorphic to a 3-star. (Contributed by AV, 13-Sep-2025.) |
⊢ 𝐺 = (5 gPetersenGr 𝐾) ⇒ ⊢ ((𝐾 ∈ (1...2) ∧ 𝑉 ∈ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑉)) ≃𝑔𝑟 (StarGr‘3)) | ||
Theorem | gpg5grlic 47974 | The two generalized Petersen graphs G(N,K) of order 10 (𝑁 = 5), which are the Petersen graph G(5,2) and the 5-prism G(5,1), are locally isomorphic. (Contributed by AV, 29-Sep-2025.) |
⊢ (5 gPetersenGr 1) ≃𝑙𝑔𝑟 (5 gPetersenGr 2) | ||
Theorem | 1hegrlfgr 47975* | A graph 𝐺 with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) & ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) & ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸) ⇒ ⊢ (𝜑 → (iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) | ||
Syntax | cupwlks 47976 | Extend class notation with walks (of a pseudograph). |
class UPWalks | ||
Definition | df-upwlks 47977* |
Define the set of all walks (in a pseudograph), called "simple walks"
in
the following.
According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)." According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4. Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Although this definition is also applicable for arbitrary hypergraphs, it allows only walks consisting of not proper hyperedges (i.e. edges connecting at most two vertices). Therefore, it should be used for pseudographs only. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ UPWalks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) | ||
Theorem | upwlksfval 47978* | The set of simple walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (UPWalks‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) | ||
Theorem | isupwlk 47979* | Properties of a pair of functions to be a simple walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | isupwlkg 47980* | Generalization of isupwlk 47979: Conditions for two classes to represent a simple walk. (Contributed by AV, 5-Nov-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | upwlkbprop 47981 | Basic properties of a simple walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 29-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(UPWalks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | ||
Theorem | upwlkwlk 47982 | A simple walk is a walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 27-Feb-2021.) |
⊢ (𝐹(UPWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | upgrwlkupwlk 47983 | In a pseudograph, a walk is a simple walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.) |
⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃) | ||
Theorem | upgrwlkupwlkb 47984 | In a pseudograph, the definitions for a walk and a simple walk are equivalent. (Contributed by AV, 30-Dec-2020.) |
⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ 𝐹(UPWalks‘𝐺)𝑃)) | ||
Theorem | upgrisupwlkALT 47985* | Alternate proof of upgriswlk 29673 using the definition of UPGraph and related theorems. (Contributed by AV, 2-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | upgredgssspr 47986 | The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021.) |
⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ (Pairs‘(Vtx‘𝐺))) | ||
Theorem | uspgropssxp 47987* | The set 𝐺 of "simple pseudographs" for a fixed set 𝑉 of vertices is a subset of a Cartesian product. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 47997. (Contributed by AV, 24-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ⊆ (𝑊 × 𝑃)) | ||
Theorem | uspgrsprfv 47988* | The value of the function 𝐹 which maps a "simple pseudograph" for a fixed set 𝑉 of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for 𝐺 as defined here, the function 𝐹 is a bijection between the "simple pseudographs" and the subsets of the set of pairs 𝑃 over the fixed set 𝑉 of vertices, see uspgrbispr 47994. (Contributed by AV, 24-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋)) | ||
Theorem | uspgrsprf 47989* | The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 24-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ 𝐹:𝐺⟶𝑃 | ||
Theorem | uspgrsprf1 47990* | The mapping 𝐹 is a one-to-one function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ 𝐹:𝐺–1-1→𝑃 | ||
Theorem | uspgrsprfo 47991* | The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 onto the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–onto→𝑃) | ||
Theorem | uspgrsprf1o 47992* | The mapping 𝐹 is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. See also the comments on uspgrbisymrel 47997. (Contributed by AV, 25-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–1-1-onto→𝑃) | ||
Theorem | uspgrex 47993* | The class 𝐺 of all "simple pseudographs" with a fixed set of vertices 𝑉 is a set. (Contributed by AV, 26-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ∈ V) | ||
Theorem | uspgrbispr 47994* | There is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 26-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑃) | ||
Theorem | uspgrspren 47995* | The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑃 of subsets of the set of pairs over the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑃) | ||
Theorem | uspgrymrelen 47996* | The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑅 of the symmetric relations on the fixed set 𝑉 are equinumerous. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 47997. (Contributed by AV, 27-Nov-2021.) |
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑅) | ||
Theorem | uspgrbisymrel 47997* |
There is a bijection between the "simple pseudographs" for a fixed
set
𝑉 of vertices and the class 𝑅 of the
symmetric relations on the
fixed set 𝑉. The simple pseudographs, which are
graphs without
hyper- or multiedges, but which may contain loops, are expressed as
ordered pairs of the vertices and the edges (as proper or improper
unordered pairs of vertices, not as indexed edges!) in this theorem.
That class 𝐺 of such simple pseudographs is a set
(if 𝑉 is a
set, see uspgrex 47993) of equivalence classes of graphs
abstracting from
the index sets of their edge functions.
Solely for this abstraction, there is a bijection between the "simple pseudographs" as members of 𝐺 and the symmetric relations 𝑅 on the fixed set 𝑉 of vertices. This theorem would not hold for 𝐺 = {𝑔 ∈ USPGraph ∣ (Vtx‘𝑔) = 𝑉} and even not for 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ 〈𝑣, 𝑒〉 ∈ USPGraph)}, because these are much bigger classes. (Proposed by Gerard Lang, 16-Nov-2021.) (Contributed by AV, 27-Nov-2021.) |
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅) | ||
Theorem | uspgrbisymrelALT 47998* | Alternate proof of uspgrbisymrel 47997 not using the definition of equinumerosity. (Contributed by AV, 26-Nov-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅) | ||
Theorem | ovn0dmfun 47999 | If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6949. (Contributed by AV, 27-Jan-2020.) |
⊢ ((𝐴𝐹𝐵) ≠ ∅ → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝐴, 𝐵〉}))) | ||
Theorem | xpsnopab 48000* | A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.) |
⊢ ({𝑋} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)} |
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