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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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