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Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrusgraprop2 26201* The properties of a k-regular undirected simple graph expressed with neighbors. (Contributed by Alexander van der Vekens, 26-Jul-2018.)
(⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑝)) = 𝐾))
 
Theoremrusgraprop3 26202* The properties of a k-regular undirected simple graph expressed with edges. (Contributed by Alexander van der Vekens, 26-Jul-2018.)
(⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 (#‘{𝑛𝑉 ∣ {𝑝, 𝑛} ∈ ran 𝐸}) = 𝐾))
 
Theoremrusgraprop4 26203* The properties of a k-regular undirected simple graph expressed with trailing edges of walks (as words). (Contributed by Alexander van der Vekens, 2-Aug-2018.)
(⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ Word 𝑉(𝑝 ≠ ∅ → (#‘{𝑛𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾)))
 
Theoremrusgrasn 26204 If a k-regular undirected simple graph has only one vertex, then k must be 0. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
(((#‘𝑉) = 1 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → 𝐾 = 0)
 
16.1.7.2  Walks in regular graphs
 
Theoremrusgranumwwlkl1 26205* In a k-regular graph, the number of walks of length 1 represented as words (thus the number of edges) starting at a fixed vertex is k. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Proof shortened by AV, 4-May-2021.)
((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}) = 𝐾)
 
Theoremrusgranumwlkl1 26206* In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∣ (𝑤‘0) = 𝑃}) = 𝐾)
 
Theoremrusgranumwlklem0 26207* Lemma 0 for rusgranumwlk 26216. (Contributed by Alexander van der Vekens, 23-Aug-2018.)
(𝑌 ∈ {𝑤𝑍 ∣ (𝑤‘0) = 𝑃} → {𝑤𝑋 ∣ (𝜑𝜓)} = {𝑤𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃𝜓)})
 
Theoremrusgranumwlklem1 26208* Lemma 1 for rusgranumwlk 26216. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})       (𝑅 ∈ (𝑊𝑁) → (𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑅)) = 𝑁))
 
Theoremrusgranumwlklem2 26209* Lemma 2 for rusgranumwlk 26216. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})    &   𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))       ((𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑊𝑁) ∣ ((2nd𝑤)‘0) = 𝑃}))
 
Theoremrusgranumwlklem3 26210* Lemma 3 for rusgranumwlk 26216. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})    &   𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))       ((𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}))
 
Theoremrusgranumwlklem4 26211* Lemma 4 for rusgranumwlk 26216. (Contributed by Alexander van der Vekens, 24-Jul-2018.) (Proof shortened by AV, 5-May-2021.)
𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})    &   𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))       ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}))
 
Theoremrusgranumwlkb0 26212* Induction base 0 for rusgranumwlk 26216. Here, we do not need the regularity of the graph yet. (Contributed by Alexander van der Vekens, 24-Jul-2018.)
𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})    &   𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))       ((𝑉 USGrph 𝐸𝑃𝑉) → (𝑃𝐿0) = 1)
 
Theoremrusgranumwlkb1 26213* Induction base 1 for rusgranumwlk 26216. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})    &   𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))       ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑃𝑉) → (𝑃𝐿1) = 𝐾)
 
Theoremrusgra0edg 26214* Special case for graphs without edges: There are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.)
𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})    &   𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))       ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0)
 
Theoremrusgranumwlks 26215* Induction step for rusgranumwlk 26216. (Contributed by Alexander van der Vekens, 24-Aug-2018.)
𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})    &   𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))       ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
 
Theoremrusgranumwlk 26216* In a k-regular graph, the number of walks of a fixed length n from a fixed vertex is k to the power of n. We denote with (𝑊𝑛) the set of walks with length n (in a given undirected simple graph) and with (𝑣𝐿𝑛) the number of walks with length n starting at the vertex v. This theorem corresponds to statement 11 in [Huneke] p. 2: "The total number of walks v(0) v(1) ... v(n-2) from a fixed vertex v = v(0) is k^(n-2) as G is k-regular.". Because of the k-regularity, the walk can be continued in k different ways at each vertex in the walk, therefore n times. This theorem even holds for n=0: then the walk consists only of one vertex v(0), so the number of walks of length n=0 starting with v=v(0) is 1=k^0. (Contributed by Alexander van der Vekens, 24-Aug-2018.)
𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑐)) = 𝑛})    &   𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊𝑛) ∣ ((2nd𝑤)‘0) = 𝑣}))       ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾𝑁))
 
Theoremrusgranumwlkg 26217* In a k-regular graph, the number of walks of a fixed length n from a fixed vertex is k to the power of n. This theorem corresponds to statement 11 in [Huneke] p. 2: "The total number of walks v(0) v(1) ... v(n-2) from a fixed vertex v = v(0) is k^(n-2) as G is k-regular.". This theorem even holds for n=0: then the walk consists only of one vertex v(0), so the number of walks of length n=0 starting with v=v(0) is 1=k^0. Closed form of rusgranumwlk 26216. (Contributed by Alexander van der Vekens, 24-Aug-2018.)
((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}) = (𝐾𝑁))
 
Theoremrusgranumwwlkg 26218* In a k-regular graph, the number of walks (represented by words) of a fixed length n from a fixed vertex is k to the power of n. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Proof shortened by AV, 5-May-2021.)
((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁))
 
Theoremclwlknclwlkdifs 26219 The set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between the set of walks of length n starting with this vertex and the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
𝐴 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}    &   𝐵 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}       𝐴 = ({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)
 
Theoremclwlknclwlkdifnum 26220* In a k-regular graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between k to the power of n and the size of the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
𝐴 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}    &   𝐵 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}       (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘𝐴) = ((𝐾𝑁) − (#‘𝐵)))
 
16.2  Eulerian paths and the Konigsberg Bridge problem
 
16.2.1  Eulerian paths
 
Syntaxceup 26221 Extend class notation with Eulerian paths.
class EulPaths
 
Definitiondf-eupa 26222* Define the set of all Eulerian paths on an undirected multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑣 UMGrph 𝑒 ∧ ∃𝑛 ∈ ℕ0 (𝑓:(1...𝑛)–1-1-onto→dom 𝑒𝑝:(0...𝑛)⟶𝑣 ∧ ∀𝑘 ∈ (1...𝑛)(𝑒‘(𝑓𝑘)) = {(𝑝‘(𝑘 − 1)), (𝑝𝑘)}))})
 
Theoremreleupa 26223 The set (𝑉 EulPaths 𝐸) of all Eulerian paths on 𝑉, 𝐸 is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.)
Rel (𝑉 EulPaths 𝐸)
 
Theoremiseupa 26224* The property "𝐹, 𝑃 is an Eulerian path on the graph 𝑉, 𝐸". An Eulerian path is defined as bijection 𝐹 from the edges to a set 1...𝑁 a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 1 ≤ 𝑘𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘 − 1) to 𝑃(𝑘). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
(dom 𝐸 = 𝐴 → (𝐹(𝑉 EulPaths 𝐸)𝑃 ↔ (𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ0 (𝐹:(1...𝑛)–1-1-onto𝐴𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))))
 
Theoremeupagra 26225 If an eulerian path exists, then 𝑉, 𝐸 is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝐹(𝑉 EulPaths 𝐸)𝑃𝑉 UMGrph 𝐸)
 
Theoremeupai 26226* Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.)
((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) → (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–1-1-onto𝐴𝑃:(0...(#‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))
 
Theoremeupatrl 26227* An Eulerian path is a trail.

Unfortunately, the edge function 𝐹 of an Eulerian path has the domain (1...(#‘𝐹)), whereas the edge functions of all kinds of walks defined here have the domain (0..^(#‘𝐹)) (i.e. the edge functions are "words of edge indices", see discussion and proposal of Mario Carneiro at https://groups.google.com/d/msg/metamath/KdVXdL3IH3k/2-BYcS_ACQAJ). Therefore, the arguments of the edge function of an Eulerian path must be shifted by 1 to obtain an edge function of a trail in this theorem, using the auxiliary theorems above (fargshiftlem 25894, fargshiftfv 25895, etc.). TODO: The definition of an Eulerian path and all related theorems should be modified to fit to the general definition of a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.)

𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))       (𝐹(𝑉 EulPaths 𝐸)𝑃𝐺(𝑉 Trails 𝐸)𝑃)
 
Theoremeupacl 26228 An Eulerian path has length #(𝐹), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝐹(𝑉 EulPaths 𝐸)𝑃 → (#‘𝐹) ∈ ℕ0)
 
Theoremeupaf1o 26229 The 𝐹 function in an Eulerian path is a bijection from a one-based sequence to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.)
((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) → 𝐹:(1...(#‘𝐹))–1-1-onto𝐴)
 
Theoremeupafi 26230 Any graph with an Eulerian path is finite. (Contributed by Mario Carneiro, 7-Apr-2015.)
((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) → 𝐴 ∈ Fin)
 
Theoremeupapf 26231 The 𝑃 function in an Eulerian path is a function from a zero-based finite sequence to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝐹(𝑉 EulPaths 𝐸)𝑃𝑃:(0...(#‘𝐹))⟶𝑉)
 
Theoremeupaseg 26232 The 𝑁-th edge in an eulerian path is the edge from 𝑃(𝑁 − 1) to 𝑃(𝑁). (Contributed by Mario Carneiro, 12-Mar-2015.)
((𝐹(𝑉 EulPaths 𝐸)𝑃𝑁 ∈ (1...(#‘𝐹))) → (𝐸‘(𝐹𝑁)) = {(𝑃‘(𝑁 − 1)), (𝑃𝑁)})
 
Theoremeupa0 26233 There is an Eulerian path on the empty graph. (Contributed by Mario Carneiro, 7-Apr-2015.)
((𝑉𝑊𝐴𝑉) → ∅(𝑉 EulPaths ∅){⟨0, 𝐴⟩})
 
Theoremeupares 26234 The restriction of an Eulerian path to an initial segment of the path forms an Eulerian path on the subgraph consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
(𝜑𝐺(𝑉 EulPaths 𝐸)𝑃)    &   (𝜑𝑁 ∈ (0...(#‘𝐺)))    &   𝐹 = (𝐸 ↾ (𝐺 “ (1...𝑁)))    &   𝐻 = (𝐺 ↾ (1...𝑁))    &   𝑄 = (𝑃 ↾ (0...𝑁))       (𝜑𝐻(𝑉 EulPaths 𝐹)𝑄)
 
Theoremeupap1 26235 Append one path segment to an Eulerian path (enlarging the graph to add the new edge). (Contributed by Mario Carneiro, 7-Apr-2015.)
(𝜑𝐸 Fn 𝐴)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶𝑉)    &   (𝜑 → ¬ 𝐵𝐴)    &   (𝜑𝐺(𝑉 EulPaths 𝐸)𝑃)    &   (𝜑𝑁 = (#‘𝐺))    &   𝐹 = (𝐸 ∪ {⟨𝐵, {(𝑃𝑁), 𝐶}⟩})    &   𝐻 = (𝐺 ∪ {⟨(𝑁 + 1), 𝐵⟩})    &   𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})       (𝜑𝐻(𝑉 EulPaths 𝐹)𝑄)
 
Theoremeupath2lem1 26236 Lemma for eupath2 26239. (Contributed by Mario Carneiro, 8-Apr-2015.)
(𝑈𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
 
Theoremeupath2lem2 26237 Lemma for eupath2 26239. (Contributed by Mario Carneiro, 8-Apr-2015.)
𝐵 ∈ V       ((𝐵𝐶𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
 
Theoremeupath2lem3 26238* Lemma for eupath2 26239. (Contributed by Mario Carneiro, 8-Apr-2015.)
(𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹(𝑉 EulPaths 𝐸)𝑃)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑁 + 1) ≤ (#‘𝐹))    &   (𝜑𝑈𝑉)    &   (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...𝑁))))‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))       (𝜑 → (¬ 2 ∥ ((𝑉 VDeg (𝐸 ↾ (𝐹 “ (1...(𝑁 + 1)))))‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))
 
Theoremeupath2 26239* The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.)
(𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹(𝑉 EulPaths 𝐸)𝑃)       (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))
 
Theoremeupath 26240* A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.)
((𝑉 EulPaths 𝐸) ≠ ∅ → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((𝑉 VDeg 𝐸)‘𝑥)}) ∈ {0, 2})
 
16.2.2  The Konigsberg Bridge problem
 
Theoremvdeg0i 26241 The base case for the induction for calculating the degree of a vertex. The degree of 𝑈 in the empty graph is 0. (Contributed by Mario Carneiro, 12-Mar-2015.)
𝑉 ∈ V    &   𝑈𝑉       ((𝑉 VDeg ∅)‘𝑈) = 0
 
Theoremumgrabi 26242* Show that an unordered pair is a valid edge in a graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝑉 ∈ V    &   𝑋𝑉    &   𝑌𝑉       (𝜑 → {𝑋, 𝑌} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
 
Theoremvdegp1ai 26243* The induction step for a vertex degree calculation. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑋, 𝑌} to the edge set, where 𝑋𝑈𝑌, yields degree 𝑃 as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝑉 ∈ V    &   (⊤ → 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})    &   𝑈𝑉    &   ((𝑉 VDeg 𝐸)‘𝑈) = 𝑃    &   𝑋𝑉    &   𝑋𝑈    &   𝑌𝑉    &   𝑌𝑈    &   𝐹 = (𝐸 ++ ⟨“{𝑋, 𝑌}”⟩)       ((𝑉 VDeg 𝐹)‘𝑈) = 𝑃
 
Theoremvdegp1bi 26244* The induction step for a vertex degree calculation. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑈, 𝑋} to the edge set, where 𝑋𝑈, yields degree 𝑃 + 1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝑉 ∈ V    &   (⊤ → 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})    &   𝑈𝑉    &   ((𝑉 VDeg 𝐸)‘𝑈) = 𝑃    &   𝑄 = (𝑃 + 1)    &   𝑋𝑉    &   𝑋𝑈    &   𝐹 = (𝐸 ++ ⟨“{𝑈, 𝑋}”⟩)       ((𝑉 VDeg 𝐹)‘𝑈) = 𝑄
 
Theoremvdegp1ci 26245* The induction step for a vertex degree calculation. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑋, 𝑈} to the edge set, where 𝑋𝑈, yields degree 𝑃 + 1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝑉 ∈ V    &   (⊤ → 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})    &   𝑈𝑉    &   ((𝑉 VDeg 𝐸)‘𝑈) = 𝑃    &   𝑄 = (𝑃 + 1)    &   𝑋𝑉    &   𝑋𝑈    &   𝐹 = (𝐸 ++ ⟨“{𝑋, 𝑈}”⟩)       ((𝑉 VDeg 𝐹)‘𝑈) = 𝑄
 
Theoremkonigsberg 26246 The Konigsberg Bridge problem. If 𝑉, 𝐸 is the graph on four vertices 0, 1, 2, 3, with edges {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 2}, {2, 3}, {2, 3}, then vertices 0, 1, 3 each have degree three, and 2 has degree five, so there are four vertices of odd degree and thus by eupath 26240 the graph cannot have an Eulerian path. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩       (𝑉 EulPaths 𝐸) = ∅
 
16.3  The Friendship Theorem

In this section, the basics for the friendship theorem, which is one from the "100 theorem list" (#83), are provided (subsection "Friendship graphs - basics"), including the definition of friendship graphs df-frgra 26248 as special undirected simple graphs without loops (see frisusgra 26251). In subsection "The friendship theorem for small graphs", the friendship theorem for small graphs (with up to 3 vertices) is proved, see 1to3vfriendship 26267. The general friendship theorem friendship 26381 (((𝑉 FriendGrph 𝐸𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸) is proven by following the approach of [Huneke] in subsection "Huneke's Proof of the Friendship Theorem". The case 𝑉 = ∅ (a graph without vertices) must be excluded either from the definition of a friendship graph, or from the theorem. If it is not excluded from the definition, which is the case with df-frgra 26248, a graph without vertices is a friendship graph (see frgra0 26253), but the friendship condition 𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸 does not hold (because of ¬ ∃𝑥 ∈ ∅𝜑, see rex0 3797).

Further results of this sections are: Any graph with exactly one vertex is a friendship graph, see frgra1v 26257, any graph with exactly 2 (different) vertices is not a friendship graph, see frgra2v 26258, a graph with exactly 3 (different) vertices is a friendship graph if and only if it is a complete graph (every two vertices are connected by an edge), see frgra3v 26261, and every friendship graph (with 1 or 3 vertices) is a windmill graph, see 1to3vfriswmgra 26266 (The generalization of this theorem "Every friendship graph (with at least one vertex) is a windmill graph" is a stronger result than the "friendship theorem". This generalization was proven by Mertzios and Unger, see Theorem 1 of [MertziosUnger] p. 152.).

In subsection "Theorems according to Mertzios and Unger", the first steps to prove the friendship theorem following the approach of Mertzios and Unger are made by 2pthfrgrarn2 26269 and n4cyclfrgra 26277 (these theorems correspond to Proposition 1 of [MertziosUnger] p. 153.).

 
16.3.1  Friendship graphs - basics
 
Syntaxcfrgra 26247 Extend class notation with Friendship Graphs.
class FriendGrph
 
Definitiondf-frgra 26248* Define the class of all Friendship Graphs. A graph is called a friendship graph if every pair of its vertices has exactly one common neighbor. (Contributed by Alexander van der Vekens and Mario Carneiro, 2-Oct-2017.)
FriendGrph = {⟨𝑣, 𝑒⟩ ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒)}
 
Theoremisfrgra 26249* The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 FriendGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
 
Theoremfrisusgrapr 26250* A friendship graph is an undirected simple graph without loops with special properties. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
(𝑉 FriendGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))
 
Theoremfrisusgra 26251 A friendship graph is an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
(𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
 
Theoremfrgra0v 26252 Any graph with no vertex is a friendship graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
(∅ FriendGrph 𝐸𝐸 = ∅)
 
Theoremfrgra0 26253 Any empty graph (graph without vertices) is a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
∅ FriendGrph ∅
 
Theoremfrgraunss 26254* Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
(𝑉 FriendGrph 𝐸 → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ ran 𝐸))
 
Theoremfrgraun 26255* Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
(𝑉 FriendGrph 𝐸 → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))
 
Theoremfrisusgranb 26256* In a friendship graph, the neighborhoods of two different vertices have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
(𝑉 FriendGrph 𝐸 → ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥𝑉 ((⟨𝑉, 𝐸⟩ Neighbors 𝑘) ∩ (⟨𝑉, 𝐸⟩ Neighbors 𝑙)) = {𝑥})
 
16.3.2  The friendship theorem for small graphs
 
Theoremfrgra1v 26257 Any graph with only one vertex is a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
((𝑉𝑋 ∧ {𝑉} USGrph 𝐸) → {𝑉} FriendGrph 𝐸)
 
Theoremfrgra2v 26258 Any graph with two (different) vertices is not a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Proof shortened by Alexander van der Vekens, 13-Sep-2018.)
(((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) → ¬ {𝐴, 𝐵} FriendGrph 𝐸)
 
Theoremfrgra3vlem1 26259* Lemma 1 for frgra3v 26261. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ∀𝑥𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸)) → 𝑥 = 𝑦))
 
Theoremfrgra3vlem2 26260* Lemma 2 for frgra3v 26261. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
 
Theoremfrgra3v 26261 Any graph with three vertices which are completely connected with each other is a friendship graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐵, 𝐶} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))))
 
Theorem1vwmgra 26262* Every graph with one vertex is a windmill graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
((𝐴𝑋𝑉 = {𝐴}) → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸))
 
Theorem3vfriswmgralem 26263* Lemma for 3vfriswmgra 26264. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
(((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸))
 
Theorem3vfriswmgra 26264* Every friendship graph with three (different) vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
 
Theorem1to2vfriswmgra 26265* Every friendship graph with one or two vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵})) → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
 
Theorem1to3vfriswmgra 26266* Every friendship graph with one, two or three vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶})) → (𝑉 FriendGrph 𝐸 → ∃𝑉𝑣 ∈ (𝑉 ∖ {})({𝑣, } ∈ ran 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {}){𝑣, 𝑤} ∈ ran 𝐸)))
 
Theorem1to3vfriendship 26267* The friendship theorem for small graphs: In every friendship graph with one, two or three vertices, there is a vertex which is adjacent to all other vertices. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
((𝐴𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶})) → (𝑉 FriendGrph 𝐸 → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
 
16.3.3  Theorems according to Mertzios and Unger
 
Theorem2pthfrgrarn 26268* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 15-Nov-2017.)
(𝑉 FriendGrph 𝐸 → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸))
 
Theorem2pthfrgrarn2 26269* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 16-Nov-2017.)
(𝑉 FriendGrph 𝐸 → ∀𝑎𝑉𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏𝑉 (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸) ∧ (𝑎𝑏𝑏𝑐)))
 
Theorem2pthfrgra 26270* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 6-Dec-2017.)
(𝑉 FriendGrph 𝐸 → ∀𝑎𝑉𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑓𝑝(𝑓(𝑎(𝑉 PathOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2))
 
Theorem3cyclfrgrarn1 26271* Every vertex in a friendship graph ( with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.)
((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐶𝑉) ∧ 𝐴𝐶) → ∃𝑏𝑉𝑐𝑉 ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝐴} ∈ ran 𝐸))
 
Theorem3cyclfrgrarn 26272* Every vertex in a friendship graph ( with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.)
((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))
 
Theorem3cyclfrgrarn2 26273* Every vertex in a friendship graph ( with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → ∀𝑎𝑉𝑏𝑉𝑐𝑉 (𝑏𝑐 ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
 
Theorem3cyclfrgra 26274* Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → ∀𝑣𝑉𝑓𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝑣))
 
Theorem4cycl2v2nb 26275 In a (maybe degenerated) 4-cycle, two vertices have two (maybe not different) common neighbors. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) → ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ ran 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ ran 𝐸))
 
Theorem4cycl2vnunb 26276* In a 4-cycle, two distinct vertices have not a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ¬ ∃!𝑥𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ ran 𝐸)
 
Theoremn4cyclfrgra 26277 There is no 4-cycle in a friendship graph, see Proposition 1(a) of [MertziosUnger] p. 153 : "A friendship graph G contains no C4 as a subgraph ...". (Contributed by Alexander van der Vekens, 19-Nov-2017.)
((𝑉 FriendGrph 𝐸𝐹(𝑉 Cycles 𝐸)𝑃) → (#‘𝐹) ≠ 4)
 
Theorem4cyclusnfrgra 26278 A graph with a 4-cycle is not a friendhip graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐶𝑉𝐴𝐶) ∧ (𝐵𝑉𝐷𝑉𝐵𝐷)) → ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐷} ∈ ran 𝐸 ∧ {𝐷, 𝐴} ∈ ran 𝐸)) → ¬ 𝑉 FriendGrph 𝐸))
 
Theoremfrgranbnb 26279 If two neighbors of a specific vertex have a common neighbor in a friendship graph, then this common neighbor must be the specific vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
(𝜑𝑋𝑉)    &   𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   (𝜑𝑉 FriendGrph 𝐸)       ((𝜑 ∧ (𝑈𝐷𝑊𝐷) ∧ 𝑈𝑊) → (({𝑈, 𝐴} ∈ ran 𝐸 ∧ {𝑊, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝑋))
 
Theoremfrconngra 26280 A friendship graph is connected, see remark 1 in [MertziosUnger] p. 153 (after Proposition 1): "An arbitrary friendship graph has to be connected, ... ". (Contributed by Alexander van der Vekens, 6-Dec-2017.)
(𝑉 FriendGrph 𝐸𝑉 ConnGrph 𝐸)
 
Theoremvdfrgra0 26281 A vertex in a friendship graph has degree 0 if the graph consists of only one vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
((𝑉 FriendGrph 𝐸𝑁𝑉 ∧ (#‘𝑉) = 1) → ((𝑉 VDeg 𝐸)‘𝑁) = 0)
 
Theoremvdn0frgrav2 26282 A vertex in a friendship graph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.)
((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 0))
 
Theoremvdgn0frgrav2 26283 A vertex in a friendship graph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
((𝑉 FriendGrph 𝐸𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 0))
 
Theoremvdn1frgrav2 26284 Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 10-Dec-2017.)
((𝑉 FriendGrph 𝐸𝐸 ∈ Fin ∧ 𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1))
 
Theoremvdgn1frgrav2 26285 Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 21-Dec-2017.)
((𝑉 FriendGrph 𝐸𝑁𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 1))
 
Theoremvdgfrgragt2 26286 Any vertex in a friendship graph (with more than one vertex - then, actually, the graph must have at least three vertices, because otherwise, it would not be a friendship graph) has at least degree 2, see remark 3 in [MertziosUnger] p. 153 (after Proposition 1): "It follows that deg(v) >= 2 for every node v of a friendship graph". (Contributed by Alexander van der Vekens, 21-Dec-2017.)
((𝑉 FriendGrph 𝐸𝑁𝑉) → (1 < (#‘𝑉) → 2 ≤ ((𝑉 VDeg 𝐸)‘𝑁)))
 
Theoremvdgn1frgrav3 26287* Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 4-Sep-2018.)
((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 1)
 
Theoremusgn0fidegnn0 26288* In a nonempty, finite graph there is a vertex having a nonnegative integer as degree. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ∃𝑣𝑉𝑛 ∈ ℕ0 ((𝑉 VDeg 𝐸)‘𝑣) = 𝑛)
 
16.3.4  Huneke's Proof of the Friendship Theorem

In this section, the friendship theorem friendship 26381 is proven by formalizing Huneke's proof, see [Huneke] pp. 1-2. The three claims (see frgrancvvdgeq 26302, frgraregorufr 26312 and frgregordn0 26329) and additional statements (numbered in the order of their occurence in the paper) in Huneke's proof are cited in the corresponding theorems.

 
Theoremfrgrancvvdeqlem1 26289* Lemma 1 for frgrancvvdeq 26301. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       ((𝜑𝑥𝐷) → 𝑌 ∈ (𝑉 ∖ {𝑥}))
 
Theoremfrgrancvvdeqlem2 26290* Lemma 2 for frgrancvvdeq 26301. (Contributed by Alexander van der Vekens, 23-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       (𝜑𝑋𝑁)
 
Theoremfrgrancvvdeqlem3 26291* Lemma 3 for frgrancvvdeq 26301. In a friendship graph, for each neighbor of a vertex there is exacly one neighbor of another vertex so that there is an edge between these two neighbors. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       ((𝜑𝑥𝐷) → ∃!𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸)
 
Theoremfrgrancvvdeqlem4 26292* Lemma 4 for frgrancvvdeq 26301. The restricted iota of a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       ((𝜑𝑥𝐷) → {(𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁))
 
Theoremfrgrancvvdeqlem5 26293* Lemma 5 for frgrancvvdeq 26301. The mapping of neighbors to neighbors is a function. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       (𝜑𝐴:𝐷𝑁)
 
Theoremfrgrancvvdeqlem6 26294* Lemma 6 for frgrancvvdeq 26301. The mapping of neighbors to neighbors applied on a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 23-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       ((𝜑𝑥𝐷) → {(𝐴𝑥)} = ((⟨𝑉, 𝐸⟩ Neighbors 𝑥) ∩ 𝑁))
 
Theoremfrgrancvvdeqlem7 26295* Lemma 7 for frgrancvvdeq 26301. (Contributed by Alexander van der Vekens, 23-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ ran 𝐸)
 
TheoremfrgrancvvdeqlemA 26296* Lemma A for frgrancvvdeq 26301. This corresponds to statement 1 in [Huneke] p. 1: "This common neighbor cannot be x, as x and y are not adjacent.". (Contributed by Alexander van der Vekens, 23-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       (𝜑 → ∀𝑥𝐷 (𝐴𝑥) ≠ 𝑋)
 
TheoremfrgrancvvdeqlemB 26297* Lemma B for frgrancvvdeq 26301. This corresponds to statement 2 in [Huneke] p. 1: "The map is one-to-one since z in N(x) is uniquely determined as the common neighbor of x and a(x)". (Contributed by Alexander van der Vekens, 23-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       (𝜑𝐴:𝐷1-1→ran 𝐴)
 
TheoremfrgrancvvdeqlemC 26298* Lemma C for frgrancvvdeq 26301. This corresponds to statement 3 in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       (𝜑𝐴:𝐷onto𝑁)
 
Theoremfrgrancvvdeqlem8 26299* Lemma 8 for frgrancvvdeq 26301. (Contributed by Alexander van der Vekens, 24-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       (𝜑𝐴:𝐷1-1-onto𝑁)
 
Theoremfrgrancvvdeqlem9 26300* Lemma 9 for frgrancvvdeq 26301. (Contributed by Alexander van der Vekens, 24-Dec-2017.)
(𝑉 FriendGrph 𝐸 → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑥)–1-1-onto→(⟨𝑉, 𝐸⟩ Neighbors 𝑦)))
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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