P. 481 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48001-48100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	grlicer 48001	Local isomorphism is an equivalence relation on hypergraphs. (Contributed by AV, 11-Jun-2025.)
⊢ ( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph
Theorem	grlicen 48002	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 48003	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	clnbgr3stgrgrlim 48004*	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 any bijection between the vertices is a local isomorphism between the two graphs. (Contributed by AV, 28-Dec-2025.)
⊢ 𝑁 ∈ ℕ0    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
Theorem	clnbgr3stgrgrlic 48005*	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 48006*	Lemma for usgrexmpl1 48007. (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 48007	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 48008	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 48009	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 48010	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 48011*	Lemma for usgrexmpl2 48012. (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 48012	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 48013	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 48014	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 48015*	Lemma for usgrexmpl2nb0 48016 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 48016	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 48017	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 48018	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 48019	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 48020	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 48021	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 48022*	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 48023	The graphs 𝐻 and 𝐺 are not isomorphic (𝐻 contains a triangle, see usgrexmpl1tri 48010, whereas 𝐺 does not, see usgrexmpl2trifr 48022. (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 48024	The graphs 𝐻 and 𝐺 are not locally isomorphic (𝐻 contains a triangle, see usgrexmpl1tri 48010, whereas 𝐺 does not, see usgrexmpl2trifr 48022. (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
21.48.15.8  Generalized Petersen graphs 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 "outside vertices", the vertices of the star polygon "inside vertices" (see A. Steimle, W. Stanton, "The isomorphism classes of the generalized Petersen graphs", Discrete Mathematics Volume 309, Issue 1, 6 January 2009, Pages 231-237: https://doi.org/10.1016/j.disc.2007.12.074). Since regular polygons are also considered as star polygons (with density 1), many theorems for "inside vertices" (with labels containing the fragment "vtx1") can be specialized for "outside vertices" (with labels containing the fragment "vtx0").
Syntax	cgpg 48025	Extend class notation with generalized Petersen graphs.
class gPetersenGr
Definition	df-gpg 48026*	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 48027*	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 48028	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 48029*	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 48030*	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	gpgiedgdmellem 48031*	Lemma for gpgiedgdmel 48034 and gpgedgel 48035. (Contributed by AV, 2-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (∃𝑥 ∈ 𝐼 (𝑌 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑌 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑌 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩}) → 𝑌 ∈ 𝒫 ({0, 1} × 𝐼)))
Theorem	gpgvtxel 48032*	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 48033	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	gpgiedgdmel 48034*	An index of edges of the generalized Petersen graph GPG(N,K). (Contributed by AV, 2-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ 𝐽) → (𝑋 ∈ dom (iEdg‘𝐺) ↔ ∃𝑥 ∈ 𝐼 (𝑋 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑋 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑋 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 𝐾) mod 𝑁)⟩})))
Theorem	gpgedgel 48035*	An edge in a generalized Petersen graph 𝐺. (Contributed by AV, 29-Aug-2025.) (Proof shortened by AV, 8-Nov-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	gpgprismgriedgdmel 48036*	An index of edges of the generalized Petersen graph GPG(N,1). (Contributed by AV, 2-Nov-2025.)
⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐺 = (𝑁 gPetersenGr 1)    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑋 ∈ dom (iEdg‘𝐺) ↔ ∃𝑥 ∈ 𝐼 (𝑋 = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ 𝑋 = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ 𝑋 = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
Theorem	gpgprismgriedgdmss 48037	A subset of the index of edges of the generalized Petersen graph GPG(N,1). (Contributed by AV, 2-Nov-2025.)
⊢ (𝑁 ∈ (ℤ≥‘3) → ({{⟨0, 0⟩, ⟨0, 1⟩}, {⟨0, 0⟩, ⟨1, 0⟩}} ∪ {{⟨1, 1⟩, ⟨0, 1⟩}, {⟨1, 1⟩, ⟨1, 0⟩}}) ⊆ dom (iEdg‘(𝑁 gPetersenGr 1)))
Theorem	gpgvtx0 48038	The outside vertices 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 48039	The inside vertices 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	opgpgvtx 48040	A vertex in a generalized Petersen graph 𝐺 as ordered pair. (Contributed by AV, 1-Oct-2025.)
⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) → (⟨𝑋, 𝑌⟩ ∈ 𝑉 ↔ ((𝑋 = 0 ∨ 𝑋 = 1) ∧ 𝑌 ∈ 𝐼)))
Theorem	gpgusgralem 48041*	Lemma for gpgusgra 48042. (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 48042	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	gpgprismgrusgra 48043	The generalized Petersen graphs G(N,1), which are the N-prisms, are simple graphs. (Contributed by AV, 31-Oct-2025.)
⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 gPetersenGr 1) ∈ USGraph)
Theorem	gpgorder 48044	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 48045	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	gpgedgvtx0 48046	The edges starting at an outside vertex 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 48047	The edges starting at an inside vertex 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 48048	The edges starting at an outside vertex 𝑋 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 48049	The edges starting at an inside vertex 𝑋 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	gpgedgiov 48050	The edges of the generalized Petersen graph GPG(N,K) between an inside and an outside vertex. (Contributed by AV, 11-Nov-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘3) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → ({⟨0, 𝑋⟩, ⟨1, 𝑌⟩} ∈ 𝐸 ↔ 𝑋 = 𝑌))
Theorem	gpgedg2ov 48051	The edges of the generalized Petersen graph GPG(N,K) between two outside vertices. (Contributed by AV, 15-Nov-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 ∈ (ℤ≥‘5) ∧ 𝐾 ∈ 𝐽) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼)) → (({⟨0, ((𝑌 − 1) mod 𝑁)⟩, ⟨0, 𝑋⟩} ∈ 𝐸 ∧ {⟨0, 𝑋⟩, ⟨0, ((𝑌 + 1) mod 𝑁)⟩} ∈ 𝐸) ↔ 𝑋 = 𝑌))
Theorem	gpgedg2iv 48052	The edges of the generalized Petersen graph GPG(N,K) between two inside vertices. (Contributed by AV, 20-Nov-2025.)
⊢ 𝐽 = (1..^(⌈‘(𝑁 / 2)))    &   ⊢ 𝐼 = (0..^𝑁)    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘5) ∧ (𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝐼) ∧ (𝐾 ∈ 𝐽 ∧ ((4 · 𝐾) mod 𝑁) ≠ 0)) → (({⟨1, ((𝑌 − 𝐾) mod 𝑁)⟩, ⟨1, 𝑋⟩} ∈ 𝐸 ∧ {⟨1, 𝑋⟩, ⟨1, ((𝑌 + 𝐾) mod 𝑁)⟩} ∈ 𝐸) ↔ 𝑋 = 𝑌))
Theorem	gpg5nbgrvtx03starlem1 48053	Lemma 1 for gpg5nbgrvtx03star 48065. (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 48054	Lemma 2 for gpg5nbgrvtx03star 48065. (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 48055	Lemma 3 for gpg5nbgrvtx03star 48065. (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 48056	Lemma 1 for gpg5nbgr3star 48066. (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 48057	Lemma 2 for gpg5nbgr3star 48066. (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 48058	Lemma 3 for gpg5nbgr3star 48066. (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 48059	The (open) neighborhood of an outside vertex 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 48060	The (open) neighborhood of an inside vertex 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 48061	In a generalized Petersen graph 𝐺, every outside vertex 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 48062	In a generalized Petersen graph 𝐺, every outside vertex has exactly three (different) neighbors. (Contributed by AV, 30-Aug-2025.) The proof of gpg3nbgrvtx0 48061 can be shortened using modmknepk 47347, but then theorem 2ltceilhalf 47313 is required which is based on an "example" ex-ceil 30410. 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 48063	In a generalized Petersen graph 𝐺, every inside vertex has exactly three (different) neighbors. (Contributed by AV, 3-Sep-2025.) (Proof shortened by AV, 22-Nov-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 48064	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 48067), 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 48065*	In a generalized Petersen graph G(N,K) of order greater than 8 (3 < 𝑁), every outside vertex has exactly three (different) neighbors, and none of these neighbors are connected by an edge (i.e., the (closed) neighborhood of every outside vertex 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 48066*	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 48074. (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 48067	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	gpg3kgrtriexlem1 48068	Lemma 1 for gpg3kgrtriex 48074. (Contributed by AV, 1-Oct-2025.)
⊢ (𝐾 ∈ ℕ → 𝐾 < (⌈‘((3 · 𝐾) / 2)))
Theorem	gpg3kgrtriexlem2 48069	Lemma 2 for gpg3kgrtriex 48074. (Contributed by AV, 1-Oct-2025.)
⊢ 𝑁 = (3 · 𝐾)    ⇒   ⊢ (𝐾 ∈ ℕ → (-𝐾 mod 𝑁) = (((𝐾 mod 𝑁) + 𝐾) mod 𝑁))
Theorem	gpg3kgrtriexlem3 48070	Lemma 3 for gpg3kgrtriex 48074. (Contributed by AV, 1-Oct-2025.)
⊢ 𝑁 = (3 · 𝐾)    ⇒   ⊢ (𝐾 ∈ ℕ → 𝑁 ∈ (ℤ≥‘3))
Theorem	gpg3kgrtriexlem4 48071	Lemma 4 for gpg3kgrtriex 48074. (Contributed by AV, 1-Oct-2025.)
⊢ 𝑁 = (3 · 𝐾)    ⇒   ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ (1..^(⌈‘(𝑁 / 2))))
Theorem	gpg3kgrtriexlem5 48072	Lemma 5 for gpg3kgrtriex 48074. (Contributed by AV, 1-Oct-2025.)
⊢ 𝑁 = (3 · 𝐾)    ⇒   ⊢ (𝐾 ∈ ℕ → (𝐾 mod 𝑁) ≠ (-𝐾 mod 𝑁))
Theorem	gpg3kgrtriexlem6 48073	Lemma 6 for gpg3kgrtriex 48074: 𝐸 is an edge in the generalized Petersen graph G(N,K) with 𝑁 = 3 · 𝐾. (Contributed by AV, 1-Oct-2025.)
⊢ 𝑁 = (3 · 𝐾)    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    &   ⊢ 𝐸 = {⟨1, (𝐾 mod 𝑁)⟩, ⟨1, (-𝐾 mod 𝑁)⟩}    ⇒   ⊢ (𝐾 ∈ ℕ → 𝐸 ∈ (Edg‘𝐺))
Theorem	gpg3kgrtriex 48074*	All generalized Petersen graphs G(N,K) with 𝑁 = 3 · 𝐾 contain triangles. (Contributed by AV, 1-Oct-2025.)
⊢ 𝑁 = (3 · 𝐾)    &   ⊢ 𝐺 = (𝑁 gPetersenGr 𝐾)    ⇒   ⊢ (𝐾 ∈ ℕ → ∃𝑡 𝑡 ∈ (GrTriangles‘𝐺))
Theorem	gpg5gricstgr3 48075	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	pglem 48076	Lemma for theorems about Petersen graphs. (Contributed by AV, 10-Nov-2025.)
⊢ 2 ∈ (1..^(⌈‘(5 / 2)))
Theorem	pgjsgr 48077	A Petersen graph is a simple graph. (Contributed by AV, 10-Nov-2025.)
⊢ (5 gPetersenGr 2) ∈ USGraph
Theorem	gpg5grlim 48078	A local isomorphism between 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). (Contributed by AV, 28-Dec-2025.)
⊢ ( I ↾ ({0, 1} × (0..^5))) ∈ ((5 gPetersenGr 1) GraphLocIso (5 gPetersenGr 2))
Theorem	gpg5grlic 48079	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.) (Proof shortened by AV, 22-Nov-2025.)
⊢ (5 gPetersenGr 1) ≃𝑙𝑔𝑟 (5 gPetersenGr 2)
Theorem	gpgprismgr4cycllem1 48080	Lemma 1 for gpgprismgr4cycl0 48091: the cycle ⟨𝑃, 𝐹⟩ consists of 4 edges (i.e., has length 4). (Contributed by AV, 1-Nov-2025.)
⊢ 𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    ⇒   ⊢ (♯‘𝐹) = 4
Theorem	gpgprismgr4cycllem2 48081	Lemma 2 for gpgprismgr4cycl0 48091: the cycle ⟨𝑃, 𝐹⟩ is proper, i.e., it has no overlapping edges. (Contributed by AV, 2-Nov-2025.)
⊢ 𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    ⇒   ⊢ Fun ◡𝐹
Theorem	gpgprismgr4cycllem3 48082*	Lemma 3 for gpgprismgr4cycl0 48091. (Contributed by AV, 5-Nov-2025.)
⊢ 𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑋 ∈ (0..^4)) → ((𝐹‘𝑋) ∈ 𝒫 ({0, 1} × (0..^𝑁)) ∧ ∃𝑥 ∈ (0..^𝑁)((𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨0, ((𝑥 + 1) mod 𝑁)⟩} ∨ (𝐹‘𝑋) = {⟨0, 𝑥⟩, ⟨1, 𝑥⟩} ∨ (𝐹‘𝑋) = {⟨1, 𝑥⟩, ⟨1, ((𝑥 + 1) mod 𝑁)⟩})))
Theorem	gpgprismgr4cycllem4 48083	Lemma 4 for gpgprismgr4cycl0 48091: the cycle ⟨𝑃, 𝐹⟩ consists of 5 vertices (the first and the last vertex are identical, see gpgprismgr4cycllem6 48085. (Contributed by AV, 1-Nov-2025.)
⊢ 𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    ⇒   ⊢ (♯‘𝑃) = 5
Theorem	gpgprismgr4cycllem5 48084	Lemma 5 for gpgprismgr4cycl0 48091. (Contributed by AV, 1-Nov-2025.)
⊢ 𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    ⇒   ⊢ 𝑃 ∈ Word V
Theorem	gpgprismgr4cycllem6 48085	Lemma 6 for gpgprismgr4cycl0 48091: the cycle ⟨𝑃, 𝐹⟩ is closed, i.e., the first and the last vertex are identical. (Contributed by AV, 1-Nov-2025.)
⊢ 𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    ⇒   ⊢ (𝑃‘0) = (𝑃‘4)
Theorem	gpgprismgr4cycllem7 48086	Lemma 7 for gpgprismgr4cycl0 48091: the cycle ⟨𝑃, 𝐹⟩ is proper, i.e., it has no overlapping vertices, except the first and the last one. (Contributed by AV, 1-Nov-2025.)
⊢ 𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    ⇒   ⊢ ((𝑋 ∈ (0..^(♯‘𝑃)) ∧ 𝑌 ∈ (1..^4)) → (𝑋 ≠ 𝑌 → (𝑃‘𝑋) ≠ (𝑃‘𝑌)))
Theorem	gpgprismgr4cycllem8 48087	Lemma 8 for gpgprismgr4cycl0 48091. (Contributed by AV, 2-Nov-2025.)
⊢ 𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    &   ⊢ 𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    &   ⊢ 𝐺 = (𝑁 gPetersenGr 1)    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝐹 ∈ Word dom (iEdg‘𝐺))
Theorem	gpgprismgr4cycllem9 48088	Lemma 9 for gpgprismgr4cycl0 48091. (Contributed by AV, 3-Nov-2025.)
⊢ 𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    &   ⊢ 𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    &   ⊢ 𝐺 = (𝑁 gPetersenGr 1)    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
Theorem	gpgprismgr4cycllem10 48089	Lemma 10 for gpgprismgr4cycl0 48091. (Contributed by AV, 5-Nov-2025.)
⊢ 𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    &   ⊢ 𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    &   ⊢ 𝐺 = (𝑁 gPetersenGr 1)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → ((iEdg‘𝐺)‘(𝐹‘𝑋)) = {(𝑃‘𝑋), (𝑃‘(𝑋 + 1))})
Theorem	gpgprismgr4cycllem11 48090	Lemma 11 for gpgprismgr4cycl0 48091. (Contributed by AV, 5-Nov-2025.)
⊢ 𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    &   ⊢ 𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    &   ⊢ 𝐺 = (𝑁 gPetersenGr 1)    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝐹(Cycles‘𝐺)𝑃)
Theorem	gpgprismgr4cycl0 48091	The generalized Petersen graphs G(N,1), which are the N-prisms, have a cycle of length 4 starting at the vertex ⟨0, 0⟩. (Contributed by AV, 5-Nov-2025.)
⊢ 𝑃 = ⟨“⟨0, 0⟩⟨0, 1⟩⟨1, 1⟩⟨1, 0⟩⟨0, 0⟩”⟩    &   ⊢ 𝐹 = ⟨“{⟨0, 0⟩, ⟨0, 1⟩} {⟨0, 1⟩, ⟨1, 1⟩} {⟨1, 1⟩, ⟨1, 0⟩} {⟨1, 0⟩, ⟨0, 0⟩}”⟩    &   ⊢ 𝐺 = (𝑁 gPetersenGr 1)    ⇒   ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝐹(Cycles‘𝐺)𝑃 ∧ (♯‘𝐹) = 4))
Theorem	gpgprismgr4cyclex 48092*	The generalized Petersen graphs G(N,1), which are the N-prisms, have (at least) one cycle of length 4. (Contributed by AV, 5-Nov-2025.)
⊢ (𝑁 ∈ (ℤ≥‘3) → ∃𝑝∃𝑓(𝑓(Cycles‘(𝑁 gPetersenGr 1))𝑝 ∧ (♯‘𝑓) = 4))
Theorem	pgnioedg1 48093	An inside and an outside vertex not adjacent in a Petersen graph. (Contributed by AV, 21-Nov-2025.)
⊢ 𝐺 = (5 gPetersenGr 2)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑦 ∈ (0..^5) → ¬ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸)
Theorem	pgnioedg2 48094	An inside and an outside vertex not adjacent in a Petersen graph. (Contributed by AV, 21-Nov-2025.)
⊢ 𝐺 = (5 gPetersenGr 2)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑦 ∈ (0..^5) → ¬ {⟨1, ((𝑦 + 2) mod 5)⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸)
Theorem	pgnioedg3 48095	An inside and an outside vertex not adjacent in a Petersen graph. (Contributed by AV, 21-Nov-2025.)
⊢ 𝐺 = (5 gPetersenGr 2)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑦 ∈ (0..^5) → ¬ {⟨1, ((𝑦 + 2) mod 5)⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸)
Theorem	pgnioedg4 48096	An inside and an outside vertex not adjacent in a Petersen graph. (Contributed by AV, 21-Nov-2025.)
⊢ 𝐺 = (5 gPetersenGr 2)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑦 ∈ (0..^5) → ¬ {⟨1, ((𝑦 − 2) mod 5)⟩, ⟨0, ((𝑦 − 1) mod 5)⟩} ∈ 𝐸)
Theorem	pgnioedg5 48097	An inside and an outside vertex not adjacent in a Petersen graph. (Contributed by AV, 21-Nov-2025.)
⊢ 𝐺 = (5 gPetersenGr 2)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑦 ∈ (0..^5) → ¬ {⟨1, ((𝑦 − 1) mod 5)⟩, ⟨0, ((𝑦 + 1) mod 5)⟩} ∈ 𝐸)
Theorem	pgnbgreunbgrlem1 48098*	Lemma 1 for pgnbgreunbgr 48110. (Contributed by AV, 15-Nov-2025.)
⊢ 𝐺 = (5 gPetersenGr 2)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ⊢ ((𝐿 = ⟨0, (((2nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐿 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝐿 = ⟨0, (((2nd ‘𝑋) − 1) mod 5)⟩) → ((𝐾 = ⟨0, (((2nd ‘𝑋) + 1) mod 5)⟩ ∨ 𝐾 = ⟨1, (2nd ‘𝑋)⟩ ∨ 𝐾 = ⟨0, (((2nd ‘𝑋) − 1) mod 5)⟩) → ((𝑋 ∈ 𝑉 ∧ 𝑋 = ⟨0, 𝑦⟩) → ((𝐾 ≠ 𝐿 ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) → (({𝐾, ⟨0, 𝑏⟩} ∈ 𝐸 ∧ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸) → 𝑋 = ⟨0, 𝑏⟩)))))
Theorem	pgnbgreunbgrlem2lem1 48099*	Lemma 1 for pgnbgreunbgrlem2 48102. (Contributed by AV, 16-Nov-2025.)
⊢ 𝐺 = (5 gPetersenGr 2)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ⊢ ((((𝐿 = ⟨1, ((𝑦 + 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸)
Theorem	pgnbgreunbgrlem2lem2 48100*	Lemma 2 for pgnbgreunbgrlem2 48102. (Contributed by AV, 16-Nov-2025.)
⊢ 𝐺 = (5 gPetersenGr 2)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑁 = (𝐺 NeighbVtx 𝑋)    ⇒   ⊢ ((((𝐿 = ⟨1, ((𝑦 − 2) mod 5)⟩ ∧ 𝐾 = ⟨0, 𝑦⟩) ∧ (𝑏 ∈ (0..^5) ∧ 𝑦 ∈ (0..^5))) ∧ {𝐾, ⟨0, 𝑏⟩} ∈ 𝐸) → ¬ {⟨0, 𝑏⟩, 𝐿} ∈ 𝐸)