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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 1conngr 27901 | A graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ ConnGraph) | ||
Theorem | conngrv2edg 27902* | A vertex in a connected graph with more than one vertex is incident with at least one edge. Formerly part of proof for vdgn0frgrv2 28002. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ ConnGraph ∧ 𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉)) → ∃𝑒 ∈ ran 𝐼 𝑁 ∈ 𝑒) | ||
Theorem | vdn0conngrumgrv2 27903 | A vertex in a connected multigraph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph) ∧ (𝑁 ∈ 𝑉 ∧ 1 < (♯‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) ≠ 0) | ||
According to Wikipedia ("Eulerian path", 9-Mar-2021, https://en.wikipedia.org/wiki/Eulerian_path): "In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. ... The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. ... A graph that has an Eulerian trail but not an Eulerian circuit is called semi-Eulerian." Correspondingly, an Eulerian path is defined as "a trail containing all edges" (see definition in [Bollobas] p. 16) in df-eupth 27905 resp. iseupth 27908. (EulerPaths‘𝐺) is the set of all Eulerian paths in graph 𝐺, see eupths 27907. An Eulerian circuit (called Euler tour in the definition in [Diestel] p. 22) is "a circuit in a graph containing all the edges" (see definition in [Bollobas] p. 16), or, with other words, a circuit which is an Eulerian path. The function mapping a graph to the set of its Eulerian paths is defined as EulerPaths in df-eupth 27905, whereas there is no explicit definition for Eulerian circuits (yet): The statement "〈𝐹, 𝑃〉 is an Eulerian circuit" is formally expressed by (𝐹(EulerPaths‘𝐺)𝑃 ∧ 𝐹(Circuits‘𝐺)𝑃). Each Eulerian path can be made an Eulerian circuit by adding an edge which connects the endpoints of the Eulerian path (see eupth2eucrct 27924). Vice versa, removing one edge from a graph with an Eulerian circuit results in a graph with an Eulerian path, see eucrct2eupth 27952. An Eulerian path does not have to be a path in the meaning of definition df-pths 27425, because it may traverse some vertices more than once. Therefore, "Eulerian trail" would be a more appropriate name. The main result of this section is (one direction of) Euler's Theorem: "A non-trivial connected graph has an Euler[ian] circuit iff each vertex has even degree." (see part 1 of theorem 12 in [Bollobas] p. 16 and theorem 1.8.1 in [Diestel] p. 22) or, expressed with Eulerian paths: "A connected graph has an Euler[ian] trail from a vertex x to a vertex y (not equal with x) iff x and y are the only vertices of odd degree." (see part 2 of theorem 12 in [Bollobas] p. 17). In eulerpath 27948, it is shown that a pseudograph with an Eulerian path has either zero or two vertices of odd degree, and eulercrct 27949 shows that a pseudograph with an Eulerian circuit has only vertices of even degree. | ||
Syntax | ceupth 27904 | Extend class notation with Eulerian paths. |
class EulerPaths | ||
Definition | df-eupth 27905* | Define the set of all Eulerian paths on an arbitrary graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ EulerPaths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))}) | ||
Theorem | releupth 27906 | The set (EulerPaths‘𝐺) of all Eulerian paths on 𝐺 is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ Rel (EulerPaths‘𝐺) | ||
Theorem | eupths 27907* | The Eulerian paths on the graph 𝐺. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (EulerPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)} | ||
Theorem | iseupth 27908 | The property "〈𝐹, 𝑃〉 is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (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.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)) | ||
Theorem | iseupthf1o 27909 | The property "〈𝐹, 𝑃〉 is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (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.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)) | ||
Theorem | eupthi 27910 | Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)) | ||
Theorem | eupthf1o 27911 | The 𝐹 function in an Eulerian path is a bijection from a half-open range of nonnegative integers to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼) | ||
Theorem | eupthfi 27912 | Any graph with an Eulerian path is of finite size, i.e. with a finite number of edges. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → dom 𝐼 ∈ Fin) | ||
Theorem | eupthseg 27913 | The 𝑁-th edge in an eulerian path is the edge having 𝑃(𝑁) and 𝑃(𝑁 + 1) as endpoints . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐹(EulerPaths‘𝐺)𝑃 ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁))) | ||
Theorem | upgriseupth 27914* | The property "〈𝐹, 𝑃〉 is an Eulerian path on the pseudograph 𝐺". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | upgreupthi 27915* | Properties of an Eulerian path in a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) | ||
Theorem | upgreupthseg 27916 | The 𝑁-th edge in an eulerian path is the edge from 𝑃(𝑁) to 𝑃(𝑁 + 1). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃 ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) | ||
Theorem | eupthcl 27917 | An Eulerian path has length ♯(𝐹), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | ||
Theorem | eupthistrl 27918 | An Eulerian path is a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.) (Revised by AV, 18-Feb-2021.) |
⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | ||
Theorem | eupthiswlk 27919 | An Eulerian path is a walk. (Contributed by AV, 6-Apr-2021.) |
⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | eupthpf 27920 | The 𝑃 function in an Eulerian path is a function from a finite sequence of nonnegative integers to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) | ||
Theorem | eupth0 27921 | There is an Eulerian path on an empty graph, i.e. a graph with at least one vertex, but without an edge. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 5-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 = ∅) → ∅(EulerPaths‘𝐺){〈0, 𝐴〉}) | ||
Theorem | eupthres 27922 | The restriction 〈𝐻, 𝑄〉 of an Eulerian path 〈𝐹, 𝑃〉 to an initial segment of the path (of length 𝑁) forms an Eulerian path on the subgraph 𝑆 consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ 𝐻 = (𝐹 prefix 𝑁) & ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) & ⊢ (Vtx‘𝑆) = 𝑉 ⇒ ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) | ||
Theorem | eupthp1 27923 | Append one path segment to an Eulerian path 〈𝐹, 𝑃〉 to become an Eulerian path 〈𝐻, 𝑄〉 of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 7-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ 𝑁 = (♯‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉}) & ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) & ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) & ⊢ (Vtx‘𝑆) = 𝑉 & ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶}) ⇒ ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) | ||
Theorem | eupth2eucrct 27924 | Append one path segment to an Eulerian path 〈𝐹, 𝑃〉 which may not be an (Eulerian) circuit to become an Eulerian circuit 〈𝐻, 𝑄〉 of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. (Contributed by AV, 11-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ 𝑁 = (♯‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉}) & ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) & ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) & ⊢ (Vtx‘𝑆) = 𝑉 & ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶}) & ⊢ (𝜑 → 𝐶 = (𝑃‘0)) ⇒ ⊢ (𝜑 → (𝐻(EulerPaths‘𝑆)𝑄 ∧ 𝐻(Circuits‘𝑆)𝑄)) | ||
Theorem | eupth2lem1 27925 | Lemma for eupth2 27946. (Contributed by Mario Carneiro, 8-Apr-2015.) |
⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵)))) | ||
Theorem | eupth2lem2 27926 | Lemma for eupth2 27946. (Contributed by Mario Carneiro, 8-Apr-2015.) |
⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐵 ≠ 𝐶 ∧ 𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶}))) | ||
Theorem | trlsegvdeglem1 27927 | Lemma for trlsegvdeg 27934. (Contributed by AV, 20-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) ⇒ ⊢ (𝜑 → ((𝑃‘𝑁) ∈ 𝑉 ∧ (𝑃‘(𝑁 + 1)) ∈ 𝑉)) | ||
Theorem | trlsegvdeglem2 27928 | Lemma for trlsegvdeg 27934. (Contributed by AV, 20-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → Fun (iEdg‘𝑋)) | ||
Theorem | trlsegvdeglem3 27929 | Lemma for trlsegvdeg 27934. (Contributed by AV, 20-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → Fun (iEdg‘𝑌)) | ||
Theorem | trlsegvdeglem4 27930 | Lemma for trlsegvdeg 27934. (Contributed by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) | ||
Theorem | trlsegvdeglem5 27931 | Lemma for trlsegvdeg 27934. (Contributed by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) | ||
Theorem | trlsegvdeglem6 27932 | Lemma for trlsegvdeg 27934. (Contributed by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin) | ||
Theorem | trlsegvdeglem7 27933 | Lemma for trlsegvdeg 27934. (Contributed by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin) | ||
Theorem | trlsegvdeg 27934 | Formerly part of proof of eupth2lem3 27943: If a trail in a graph 𝐺 induces a subgraph 𝑍 with the vertices 𝑉 of 𝐺 and the edges being the edges of the walk, and a subgraph 𝑋 with the vertices 𝑉 of 𝐺 and the edges being the edges of the walk except the last one, and a subgraph 𝑌 with the vertices 𝑉 of 𝐺 and one edges being the last edge of the walk, then the vertex degree of any vertex 𝑈 of 𝐺 within 𝑍 is the sum of the vertex degree of 𝑈 within 𝑋 and the vertex degree of 𝑈 within 𝑌. Note that this theorem would not hold for arbitrary walks (if the last edge was identical with a previous edge, the degree of the vertices incident with this edge would not be increased because of this edge). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈))) | ||
Theorem | eupth2lem3lem1 27935 | Lemma for eupth2lem3 27943. (Contributed by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ0) | ||
Theorem | eupth2lem3lem2 27936 | Lemma for eupth2lem3 27943. (Contributed by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) ⇒ ⊢ (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℕ0) | ||
Theorem | eupth2lem3lem3 27937* | Lemma for eupth2lem3 27943, formerly part of proof of eupth2lem3 27943: If a loop {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} is added to a trail, the degree of the vertices with odd degree remains odd (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 21-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) & ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) & ⊢ (𝜑 → if-((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}, {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁)))) ⇒ ⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) | ||
Theorem | eupth2lem3lem4 27938* | Lemma for eupth2lem3 27943, formerly part of proof of eupth2lem3 27943: If an edge (not a loop) is added to a trail, the degree of the end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) & ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) & ⊢ (𝜑 → if-((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}, {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁)))) & ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) ∈ 𝒫 𝑉) ⇒ ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 = (𝑃‘𝑁) ∨ 𝑈 = (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) | ||
Theorem | eupth2lem3lem5 27939* | Lemma for eupth2 27946. (Contributed by AV, 25-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) & ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) & ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) ⇒ ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) ∈ 𝒫 𝑉) | ||
Theorem | eupth2lem3lem6 27940* | Formerly part of proof of eupth2lem3 27943: If an edge (not a loop) is added to a trail, the degree of vertices not being end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). Remark: This seems to be not valid for hyperedges joining more vertices than (𝑃‘0) and (𝑃‘𝑁): if there is a third vertex in the edge, and this vertex is already contained in the trail, then the degree of this vertex could be affected by this edge! (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) & ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) & ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) ⇒ ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) | ||
Theorem | eupth2lem3lem7 27941* | Lemma for eupth2lem3 27943: Combining trlsegvdeg 27934, eupth2lem3lem3 27937, eupth2lem3lem4 27938 and eupth2lem3lem6 27940. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 27-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) & ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) & ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) & ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) & ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))}) ⇒ ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑍)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) | ||
Theorem | eupthvdres 27942 | Formerly part of proof of eupth2 27946: The vertex degree remains the same for all vertices if the edges are restricted to the edges of an Eulerian path. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ 𝐻 = 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))〉 ⇒ ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺)) | ||
Theorem | eupth2lem3 27943* | Lemma for eupth2 27946. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ UPGraph) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ 𝐻 = 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))〉 & ⊢ 𝑋 = 〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))〉 & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → (𝑁 + 1) ≤ (♯‘𝐹)) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐻)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)})) ⇒ ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))}))) | ||
Theorem | eupth2lemb 27944* | Lemma for eupth2 27946 (induction basis): There are no vertices of odd degree in an Eulerian path of length 0, having no edge and identical endpoints (the single vertex of the Eulerian path). Formerly part of proof for eupth2 27946. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ UPGraph) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))〉)‘𝑥)} = ∅) | ||
Theorem | eupth2lems 27945* | Lemma for eupth2 27946 (induction step): The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct, if the Eulerian path shortened by one edge has this property. Formerly part of proof for eupth2 27946. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ UPGraph) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) ⇒ ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))〉)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)})) → ((𝑛 + 1) ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘〈𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))〉)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))})))) | ||
Theorem | eupth2 27946* | 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.) (Revised by AV, 26-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ UPGraph) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))})) | ||
Theorem | eulerpathpr 27947* | A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) | ||
Theorem | eulerpath 27948* | A pseudograph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ (EulerPaths‘𝐺) ≠ ∅) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) | ||
Theorem | eulercrct 27949* | A pseudograph with an Eulerian circuit 〈𝐹, 𝑃〉 (an "Eulerian pseudograph") has only vertices of even degree. (Contributed by AV, 12-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃 ∧ 𝐹(Circuits‘𝐺)𝑃) → ∀𝑥 ∈ 𝑉 2 ∥ ((VtxDeg‘𝐺)‘𝑥)) | ||
Theorem | eucrctshift 27950* | Cyclically shifting the indices of an Eulerian circuit 〈𝐹, 𝑃〉 results in an Eulerian circuit 〈𝐻, 𝑄〉. (Contributed by AV, 15-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) & ⊢ 𝑁 = (♯‘𝐹) & ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁)) & ⊢ 𝐻 = (𝐹 cyclShift 𝑆) & ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) ⇒ ⊢ (𝜑 → (𝐻(EulerPaths‘𝐺)𝑄 ∧ 𝐻(Circuits‘𝐺)𝑄)) | ||
Theorem | eucrct2eupth1 27951 | Removing one edge (𝐼‘(𝐹‘𝑁)) from a nonempty graph 𝐺 with an Eulerian circuit 〈𝐹, 𝑃〉 results in a graph 𝑆 with an Eulerian path 〈𝐻, 𝑄〉. This is the special case of eucrct2eupth 27952 (with 𝐽 = (𝑁 − 1)) where the last segment/edge of the circuit is removed. (Contributed by AV, 11-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) & ⊢ (Vtx‘𝑆) = 𝑉 & ⊢ (𝜑 → 0 < (♯‘𝐹)) & ⊢ (𝜑 → 𝑁 = ((♯‘𝐹) − 1)) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ 𝐻 = (𝐹 prefix 𝑁) & ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) ⇒ ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) | ||
Theorem | eucrct2eupth 27952* | Removing one edge (𝐼‘(𝐹‘𝐽)) from a graph 𝐺 with an Eulerian circuit 〈𝐹, 𝑃〉 results in a graph 𝑆 with an Eulerian path 〈𝐻, 𝑄〉. (Contributed by AV, 17-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) & ⊢ (Vtx‘𝑆) = 𝑉 & ⊢ (𝜑 → 𝑁 = (♯‘𝐹)) & ⊢ (𝜑 → 𝐽 ∈ (0..^𝑁)) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ ((0..^𝑁) ∖ {𝐽})))) & ⊢ 𝐾 = (𝐽 + 1) & ⊢ 𝐻 = ((𝐹 cyclShift 𝐾) prefix (𝑁 − 1)) & ⊢ 𝑄 = (𝑥 ∈ (0..^𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) ⇒ ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) | ||
According to Wikipedia ("Seven Bridges of Königsberg", 9-Mar-2021, https://en.wikipedia.org/wiki/Seven_Bridges_of_Koenigsberg): "The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology. The city of Königsberg in [East] Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands - Kneiphof and Lomse - which were connected to each other, or to the two mainland portions of the city, by seven bridges. The problem was to devise a walk through the city that would cross each of those bridges once and only once.". Euler proved that the problem has no solution by applying Euler's theorem to the Königsberg graph, which is obtained by replacing each land mass with an abstract "vertex" or node, and each bridge with an abstract connection, an "edge", which connects two land masses/vertices. The Königsberg graph 𝐺 is a multigraph consisting of 4 vertices and 7 edges, represented by the following ordered pair: 𝐺 = 〈(0...3), 〈“{0, 1}{0, 2} {0, 3}{1, 2}{1, 2}{2, 3}{2, 3}”〉〉, see konigsbergumgr 27958. konigsberg 27964 shows that the Königsberg graph has no Eulerian path, thus the Königsberg Bridge problem has no solution. | ||
Theorem | konigsbergvtx 27953 | The set of vertices of the Königsberg graph 𝐺. (Contributed by AV, 28-Feb-2021.) |
⊢ 𝑉 = (0...3) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (Vtx‘𝐺) = (0...3) | ||
Theorem | konigsbergiedg 27954 | The indexed edges of the Königsberg graph 𝐺. (Contributed by AV, 28-Feb-2021.) |
⊢ 𝑉 = (0...3) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (iEdg‘𝐺) = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 | ||
Theorem | konigsbergiedgw 27955* | The indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.) |
⊢ 𝑉 = (0...3) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} | ||
Theorem | konigsbergssiedgwpr 27956* | Each subset of the indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.) |
⊢ 𝑉 = (0...3) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V ∧ 𝐸 = (𝐴 ++ 𝐵)) → 𝐴 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) | ||
Theorem | konigsbergssiedgw 27957* | Each subset of the indexed edges of the Königsberg graph 𝐺 is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.) |
⊢ 𝑉 = (0...3) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V ∧ 𝐸 = (𝐴 ++ 𝐵)) → 𝐴 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | ||
Theorem | konigsbergumgr 27958 | The Königsberg graph 𝐺 is a multigraph. (Contributed by AV, 28-Feb-2021.) (Revised by AV, 9-Mar-2021.) |
⊢ 𝑉 = (0...3) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ 𝐺 ∈ UMGraph | ||
Theorem | konigsberglem1 27959 | Lemma 1 for konigsberg 27964: Vertex 0 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.) |
⊢ 𝑉 = (0...3) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ ((VtxDeg‘𝐺)‘0) = 3 | ||
Theorem | konigsberglem2 27960 | Lemma 2 for konigsberg 27964: Vertex 1 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.) |
⊢ 𝑉 = (0...3) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ ((VtxDeg‘𝐺)‘1) = 3 | ||
Theorem | konigsberglem3 27961 | Lemma 3 for konigsberg 27964: Vertex 3 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.) |
⊢ 𝑉 = (0...3) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ ((VtxDeg‘𝐺)‘3) = 3 | ||
Theorem | konigsberglem4 27962* | Lemma 4 for konigsberg 27964: Vertices 0, 1, 3 are vertices of odd degree. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.) |
⊢ 𝑉 = (0...3) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ {0, 1, 3} ⊆ {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} | ||
Theorem | konigsberglem5 27963* | Lemma 5 for konigsberg 27964: The set of vertices of odd degree is greater than 2. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.) |
⊢ 𝑉 = (0...3) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) | ||
Theorem | konigsberg 27964 | The Königsberg Bridge problem. If 𝐺 is the Königsberg graph, i.e. a 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 eulerpath 27948 the graph cannot have an Eulerian path. It is sufficient to show that there are 3 vertices of odd degree, since a graph having an Eulerian path can only have 0 or 2 vertices of odd degree. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 9-Mar-2021.) |
⊢ 𝑉 = (0...3) & ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (EulerPaths‘𝐺) = ∅ | ||
Syntax | cfrgr 27965 | Extend class notation with friendship graphs. |
class FriendGraph | ||
Definition | df-frgr 27966* | Define the class of all friendship graphs: a simple graph is called a friendship graph if every pair of its vertices has exactly one common neighbor. This condition is called the friendship condition , see definition in [MertziosUnger] p. 152. (Contributed by Alexander van der Vekens and Mario Carneiro, 2-Oct-2017.) (Revised by AV, 29-Mar-2021.) (Revised by AV, 3-Jan-2024.) |
⊢ FriendGraph = {𝑔 ∈ USGraph ∣ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒} | ||
Theorem | isfrgr 27967* | The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) (Revised by AV, 3-Jan-2024.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥 ∈ 𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸)) | ||
Theorem | frgrusgr 27968 | A friendship graph is a simple graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) |
⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph) | ||
Theorem | frgr0v 27969 | Any null graph (set with no vertices) is a friendship graph iff its edge function is empty. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ FriendGraph ↔ (iEdg‘𝐺) = ∅)) | ||
Theorem | frgr0vb 27970 | Any null graph (without vertices and edges) is a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Revised by AV, 29-Mar-2021.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ FriendGraph ) | ||
Theorem | frgruhgr0v 27971 | Any null graph (without vertices) represented as hypergraph is a friendship graph. (Contributed by AV, 29-Mar-2021.) |
⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ FriendGraph ) | ||
Theorem | frgr0 27972 | The null graph (graph without vertices) is a friendship graph. (Contributed by AV, 29-Mar-2021.) |
⊢ ∅ ∈ FriendGraph | ||
Theorem | frcond1 27973* | The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)) | ||
Theorem | frcond2 27974* | The friendship condition: any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))) | ||
Theorem | frgreu 27975* | Variant of frcond2 27974: Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 4-Jan-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))) | ||
Theorem | frcond3 27976* | The friendship condition, expressed by neighborhoods: in a friendship graph, the neighborhood of a vertex and the neighborhood of a second, different vertex have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 30-Dec-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})) | ||
Theorem | frcond4 27977* | The friendship condition, alternatively expressed by neighborhoods: 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.) (Revised by AV, 29-Mar-2021.) (Proof shortened by AV, 30-Dec-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}) | ||
Theorem | frgr1v 27978 | Any graph with (at most) one vertex is a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) |
⊢ ((𝐺 ∈ USGraph ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ FriendGraph ) | ||
Theorem | nfrgr2v 27979 | 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.) (Revised by AV, 29-Mar-2021.) |
⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph ) | ||
Theorem | frgr3vlem1 27980* | Lemma 1 for frgr3v 27982. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ∀𝑥∀𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦)) | ||
Theorem | frgr3vlem2 27981* | Lemma 2 for frgr3v 27982. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))) | ||
Theorem | frgr3v 27982 | 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.) (Revised by AV, 29-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → (𝐺 ∈ FriendGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))) | ||
Theorem | 1vwmgr 27983* | Every graph with one vertex (which may be connect with itself by (multiple) loops!) is a windmill graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Revised by AV, 31-Mar-2021.) |
⊢ ((𝐴 ∈ 𝑋 ∧ 𝑉 = {𝐴}) → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸)) | ||
Theorem | 3vfriswmgrlem 27984* | Lemma for 3vfriswmgr 27985. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐴, 𝐵} ∈ 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ 𝐸)) | ||
Theorem | 3vfriswmgr 27985* | Every friendship graph with three (different) vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ 𝑉 = {𝐴, 𝐵, 𝐶}) → (𝐺 ∈ FriendGraph → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))) | ||
Theorem | 1to2vfriswmgr 27986* | Every friendship graph with one or two vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵})) → (𝐺 ∈ FriendGraph → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))) | ||
Theorem | 1to3vfriswmgr 27987* | Every friendship graph with one, two or three vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶})) → (𝐺 ∈ FriendGraph → ∃ℎ ∈ 𝑉 ∀𝑣 ∈ (𝑉 ∖ {ℎ})({𝑣, ℎ} ∈ 𝐸 ∧ ∃!𝑤 ∈ (𝑉 ∖ {ℎ}){𝑣, 𝑤} ∈ 𝐸))) | ||
Theorem | 1to3vfriendship 27988* | 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.) (Revised by AV, 31-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ (𝑉 = {𝐴} ∨ 𝑉 = {𝐴, 𝐵} ∨ 𝑉 = {𝐴, 𝐵, 𝐶})) → (𝐺 ∈ FriendGraph → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) | ||
Theorem | 2pthfrgrrn 27989* | 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.) (Revised by AV, 1-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸)) | ||
Theorem | 2pthfrgrrn2 27990* | 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.) (Revised by AV, 1-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ (𝑉 ∖ {𝑎})∃𝑏 ∈ 𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ∧ (𝑎 ≠ 𝑏 ∧ 𝑏 ≠ 𝑐))) | ||
Theorem | 2pthfrgr 27991* | Between any two (different) vertices in a friendship graph, tere is a 2-path (simple 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.) (Revised by AV, 1-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ (𝑉 ∖ {𝑎})∃𝑓∃𝑝(𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑝 ∧ (♯‘𝑓) = 2)) | ||
Theorem | 3cyclfrgrrn1 27992* | 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.) (Revised by AV, 2-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝐴} ∈ 𝐸)) | ||
Theorem | 3cyclfrgrrn 27993* | 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.) (Revised by AV, 2-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) | ||
Theorem | 3cyclfrgrrn2 27994* | 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.) (Revised by AV, 2-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑏 ≠ 𝑐 ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))) | ||
Theorem | 3cyclfrgr 27995* | 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.) (Revised by AV, 2-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝑉)) → ∀𝑣 ∈ 𝑉 ∃𝑓∃𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3 ∧ (𝑝‘0) = 𝑣)) | ||
Theorem | 4cycl2v2nb 27996 | In a (maybe degenerate) 4-cycle, two vertice have two (maybe not different) common neighbors. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.) |
⊢ ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → ({{𝐴, 𝐵}, {𝐵, 𝐶}} ⊆ 𝐸 ∧ {{𝐴, 𝐷}, {𝐷, 𝐶}} ⊆ 𝐸)) | ||
Theorem | 4cycl2vnunb 27997* | In a 4-cycle, two distinct vertices have not a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Nov-2017.) (Revised by AV, 2-Apr-2021.) |
⊢ ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → ¬ ∃!𝑥 ∈ 𝑉 {{𝐴, 𝑥}, {𝑥, 𝐶}} ⊆ 𝐸) | ||
Theorem | n4cyclfrgr 27998 | 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.) (Revised by AV, 2-Apr-2021.) |
⊢ ((𝐺 ∈ FriendGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → (♯‘𝐹) ≠ 4) | ||
Theorem | 4cyclusnfrgr 27999 | A graph with a 4-cycle is not a friendhip graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 2-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷)) → ((({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) ∧ ({𝐶, 𝐷} ∈ 𝐸 ∧ {𝐷, 𝐴} ∈ 𝐸)) → 𝐺 ∉ FriendGraph )) | ||
Theorem | frgrnbnb 28000 | If two neighbors 𝑈 and 𝑊 of a vertex 𝑋 have a common neighbor 𝐴 in a friendship graph, then this common neighbor 𝐴 must be the vertex 𝑋. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 2-Apr-2021.) (Proof shortened by AV, 13-Feb-2022.) |
⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷) ∧ 𝑈 ≠ 𝑊) → (({𝑈, 𝐴} ∈ 𝐸 ∧ {𝑊, 𝐴} ∈ 𝐸) → 𝐴 = 𝑋)) |
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