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Type | Label | Description |
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Statement | ||
Theorem | umgr3v3e3cycl 29701* | If and only if there is a 3-cycle in a multigraph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.) (Revised by AV, 12-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β UMGraph β (βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 3) β βπ β π βπ β π βπ β π ({π, π} β πΈ β§ {π, π} β πΈ β§ {π, π} β πΈ))) | ||
Theorem | upgr4cycl4dv4e 29702* | If there is a cycle of length 4 in a pseudograph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 13-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(CyclesβπΊ)π β§ (β―βπΉ) = 4) β βπ β π βπ β π βπ β π βπ β π ((({π, π} β πΈ β§ {π, π} β πΈ) β§ ({π, π} β πΈ β§ {π, π} β πΈ)) β§ ((π β π β§ π β π β§ π β π) β§ (π β π β§ π β π β§ π β π)))) | ||
Syntax | cconngr 29703 | Extend class notation with connected graphs. |
class ConnGraph | ||
Definition | df-conngr 29704* | Define the class of all connected graphs. A graph is called connected if there is a path between every pair of (distinct) vertices. The distinctness of the vertices is not necessary for the definition, because there is always a path (of length 0) from a vertex to itself, see 0pthonv 29646 and dfconngr1 29705. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ ConnGraph = {π β£ [(Vtxβπ) / π£]βπ β π£ βπ β π£ βπβπ π(π(PathsOnβπ)π)π} | ||
Theorem | dfconngr1 29705* | Alternative definition of the class of all connected graphs, requiring paths between distinct vertices. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ ConnGraph = {π β£ [(Vtxβπ) / π£]βπ β π£ βπ β (π£ β {π})βπβπ π(π(PathsOnβπ)π)π} | ||
Theorem | isconngr 29706* | The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ π = (VtxβπΊ) β β’ (πΊ β π β (πΊ β ConnGraph β βπ β π βπ β π βπβπ π(π(PathsOnβπΊ)π)π)) | ||
Theorem | isconngr1 29707* | The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ π = (VtxβπΊ) β β’ (πΊ β π β (πΊ β ConnGraph β βπ β π βπ β (π β {π})βπβπ π(π(PathsOnβπΊ)π)π)) | ||
Theorem | cusconngr 29708 | A complete hypergraph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ ((πΊ β UHGraph β§ πΊ β ComplGraph) β πΊ β ConnGraph) | ||
Theorem | 0conngr 29709 | A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ β β ConnGraph | ||
Theorem | 0vconngr 29710 | A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ ((πΊ β π β§ (VtxβπΊ) = β ) β πΊ β ConnGraph) | ||
Theorem | 1conngr 29711 | A graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ ((πΊ β π β§ (VtxβπΊ) = {π}) β πΊ β ConnGraph) | ||
Theorem | conngrv2edg 29712* | A vertex in a connected graph with more than one vertex is incident with at least one edge. Formerly part of proof for vdgn0frgrv2 29812. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β ConnGraph β§ π β π β§ 1 < (β―βπ)) β βπ β ran πΌ π β π) | ||
Theorem | vdn0conngrumgrv2 29713 | 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.) |
β’ π = (VtxβπΊ) β β’ (((πΊ β ConnGraph β§ πΊ β UMGraph) β§ (π β π β§ 1 < (β―βπ))) β ((VtxDegβπΊ)βπ) β 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 29715 resp. iseupth 29718. (EulerPathsβπΊ) is the set of all Eulerian paths in graph πΊ, see eupths 29717. 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 29715, whereas there is no explicit definition for Eulerian circuits (yet): The statement "β¨πΉ, πβ© is an Eulerian circuit" is formally expressed by (πΉ(EulerPathsβπΊ)π β§ πΉ(CircuitsβπΊ)π). Each Eulerian path can be made an Eulerian circuit by adding an edge which connects the endpoints of the Eulerian path (see eupth2eucrct 29734). Vice versa, removing one edge from a graph with an Eulerian circuit results in a graph with an Eulerian path, see eucrct2eupth 29762. An Eulerian path does not have to be a path in the meaning of definition df-pths 29237, 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 29758, it is shown that a pseudograph with an Eulerian path has either zero or two vertices of odd degree, and eulercrct 29759 shows that a pseudograph with an Eulerian circuit has only vertices of even degree. | ||
Syntax | ceupth 29714 | Extend class notation with Eulerian paths. |
class EulerPaths | ||
Definition | df-eupth 29715* | Define the set of all Eulerian paths on an arbitrary graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
β’ EulerPaths = (π β V β¦ {β¨π, πβ© β£ (π(Trailsβπ)π β§ π:(0..^(β―βπ))βontoβdom (iEdgβπ))}) | ||
Theorem | releupth 29716 | The set (EulerPathsβπΊ) of all Eulerian paths on πΊ 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.) |
β’ Rel (EulerPathsβπΊ) | ||
Theorem | eupths 29717* | The Eulerian paths on the graph πΊ. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ (EulerPathsβπΊ) = {β¨π, πβ© β£ (π(TrailsβπΊ)π β§ π:(0..^(β―βπ))βontoβdom πΌ)} | ||
Theorem | iseupth 29718 | The property "β¨πΉ, πβ© is an Eulerian path on the graph πΊ". An Eulerian path is defined as bijection πΉ from the edges to a set 0...(π β 1) and a function π:(0...π)βΆπ into the vertices such that for each 0 β€ π < π, πΉ(π) is an edge from π(π) to π(π + 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.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(EulerPathsβπΊ)π β (πΉ(TrailsβπΊ)π β§ πΉ:(0..^(β―βπΉ))βontoβdom πΌ)) | ||
Theorem | iseupthf1o 29719 | The property "β¨πΉ, πβ© is an Eulerian path on the graph πΊ". An Eulerian path is defined as bijection πΉ from the edges to a set 0...(π β 1) and a function π:(0...π)βΆπ into the vertices such that for each 0 β€ π < π, πΉ(π) is an edge from π(π) to π(π + 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.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(EulerPathsβπΊ)π β (πΉ(WalksβπΊ)π β§ πΉ:(0..^(β―βπΉ))β1-1-ontoβdom πΌ)) | ||
Theorem | eupthi 29720 | 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.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(EulerPathsβπΊ)π β (πΉ(WalksβπΊ)π β§ πΉ:(0..^(β―βπΉ))β1-1-ontoβdom πΌ)) | ||
Theorem | eupthf1o 29721 | The πΉ 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.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(EulerPathsβπΊ)π β πΉ:(0..^(β―βπΉ))β1-1-ontoβdom πΌ) | ||
Theorem | eupthfi 29722 | 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.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(EulerPathsβπΊ)π β dom πΌ β Fin) | ||
Theorem | eupthseg 29723 | The π-th edge in an eulerian path is the edge having π(π) and π(π + 1) as endpoints . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ ((πΉ(EulerPathsβπΊ)π β§ π β (0..^(β―βπΉ))) β {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ))) | ||
Theorem | upgriseupth 29724* | The property "β¨πΉ, πβ© is an Eulerian path on the pseudograph πΊ". (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.) |
β’ πΌ = (iEdgβπΊ) & β’ π = (VtxβπΊ) β β’ (πΊ β UPGraph β (πΉ(EulerPathsβπΊ)π β (πΉ:(0..^(β―βπΉ))β1-1-ontoβdom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}))) | ||
Theorem | upgreupthi 29725* | 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.) |
β’ πΌ = (iEdgβπΊ) & β’ π = (VtxβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β (πΉ:(0..^(β―βπΉ))β1-1-ontoβdom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))})) | ||
Theorem | upgreupthseg 29726 | The π-th edge in an eulerian path is the edge from π(π) to π(π + 1). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π β§ π β (0..^(β―βπΉ))) β (πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}) | ||
Theorem | eupthcl 29727 | An Eulerian path has length β―(πΉ), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
β’ (πΉ(EulerPathsβπΊ)π β (β―βπΉ) β β0) | ||
Theorem | eupthistrl 29728 | An Eulerian path is a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.) (Revised by AV, 18-Feb-2021.) |
β’ (πΉ(EulerPathsβπΊ)π β πΉ(TrailsβπΊ)π) | ||
Theorem | eupthiswlk 29729 | An Eulerian path is a walk. (Contributed by AV, 6-Apr-2021.) |
β’ (πΉ(EulerPathsβπΊ)π β πΉ(WalksβπΊ)π) | ||
Theorem | eupthpf 29730 | The π 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.) |
β’ (πΉ(EulerPathsβπΊ)π β π:(0...(β―βπΉ))βΆ(VtxβπΊ)) | ||
Theorem | eupth0 29731 | 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ ((π΄ β π β§ πΌ = β ) β β (EulerPathsβπΊ){β¨0, π΄β©}) | ||
Theorem | eupthres 29732 | The restriction β¨π», πβ© of an Eulerian path β¨πΉ, πβ© to an initial segment of the path (of length π) forms an Eulerian path on the subgraph π 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ π» = (πΉ prefix π) & β’ π = (π βΎ (0...π)) & β’ (Vtxβπ) = π β β’ (π β π»(EulerPathsβπ)π) | ||
Theorem | eupthp1 29733 | Append one path segment to an Eulerian path β¨πΉ, πβ© to become an Eulerian path β¨π», πβ© of the supergraph π obtained by adding the new edge to the graph πΊ. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 7-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) (Revised by AV, 8-Apr-2024.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β πΌ β Fin) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β Β¬ π΅ β dom πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ π = (β―βπΉ) & β’ (π β πΈ β (EdgβπΊ)) & β’ (π β {(πβπ), πΆ} β πΈ) & β’ (iEdgβπ) = (πΌ βͺ {β¨π΅, πΈβ©}) & β’ π» = (πΉ βͺ {β¨π, π΅β©}) & β’ π = (π βͺ {β¨(π + 1), πΆβ©}) & β’ (Vtxβπ) = π & β’ ((π β§ πΆ = (πβπ)) β πΈ = {πΆ}) β β’ (π β π»(EulerPathsβπ)π) | ||
Theorem | eupth2eucrct 29734 | Append one path segment to an Eulerian path β¨πΉ, πβ© which may not be an (Eulerian) circuit to become an Eulerian circuit β¨π», πβ© of the supergraph π obtained by adding the new edge to the graph πΊ. (Contributed by AV, 11-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) (Revised by AV, 8-Apr-2024.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β πΌ β Fin) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β Β¬ π΅ β dom πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ π = (β―βπΉ) & β’ (π β πΈ β (EdgβπΊ)) & β’ (π β {(πβπ), πΆ} β πΈ) & β’ (iEdgβπ) = (πΌ βͺ {β¨π΅, πΈβ©}) & β’ π» = (πΉ βͺ {β¨π, π΅β©}) & β’ π = (π βͺ {β¨(π + 1), πΆβ©}) & β’ (Vtxβπ) = π & β’ ((π β§ πΆ = (πβπ)) β πΈ = {πΆ}) & β’ (π β πΆ = (πβ0)) β β’ (π β (π»(EulerPathsβπ)π β§ π»(Circuitsβπ)π)) | ||
Theorem | eupth2lem1 29735 | Lemma for eupth2 29756. (Contributed by Mario Carneiro, 8-Apr-2015.) |
β’ (π β π β (π β if(π΄ = π΅, β , {π΄, π΅}) β (π΄ β π΅ β§ (π = π΄ β¨ π = π΅)))) | ||
Theorem | eupth2lem2 29736 | Lemma for eupth2 29756. (Contributed by Mario Carneiro, 8-Apr-2015.) |
β’ π΅ β V β β’ ((π΅ β πΆ β§ π΅ = π) β (Β¬ π β if(π΄ = π΅, β , {π΄, π΅}) β π β if(π΄ = πΆ, β , {π΄, πΆ}))) | ||
Theorem | trlsegvdeglem1 29737 | Lemma for trlsegvdeg 29744. (Contributed by AV, 20-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) β β’ (π β ((πβπ) β π β§ (πβ(π + 1)) β π)) | ||
Theorem | trlsegvdeglem2 29738 | Lemma for trlsegvdeg 29744. (Contributed by AV, 20-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β Fun (iEdgβπ)) | ||
Theorem | trlsegvdeglem3 29739 | Lemma for trlsegvdeg 29744. (Contributed by AV, 20-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β Fun (iEdgβπ)) | ||
Theorem | trlsegvdeglem4 29740 | Lemma for trlsegvdeg 29744. (Contributed by AV, 21-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β dom (iEdgβπ) = ((πΉ β (0..^π)) β© dom πΌ)) | ||
Theorem | trlsegvdeglem5 29741 | Lemma for trlsegvdeg 29744. (Contributed by AV, 21-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β dom (iEdgβπ) = {(πΉβπ)}) | ||
Theorem | trlsegvdeglem6 29742 | Lemma for trlsegvdeg 29744. (Contributed by AV, 21-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β dom (iEdgβπ) β Fin) | ||
Theorem | trlsegvdeglem7 29743 | Lemma for trlsegvdeg 29744. (Contributed by AV, 21-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β dom (iEdgβπ) β Fin) | ||
Theorem | trlsegvdeg 29744 | Formerly part of proof of eupth2lem3 29753: If a trail in a graph πΊ induces a subgraph π with the vertices π of πΊ and the edges being the edges of the walk, and a subgraph π with the vertices π of πΊ and the edges being the edges of the walk except the last one, and a subgraph π with the vertices π of πΊ and one edges being the last edge of the walk, then the vertex degree of any vertex π of πΊ within π is the sum of the vertex degree of π within π and the vertex degree of π within π. 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β ((VtxDegβπ)βπ) = (((VtxDegβπ)βπ) + ((VtxDegβπ)βπ))) | ||
Theorem | eupth2lem3lem1 29745 | Lemma for eupth2lem3 29753. (Contributed by AV, 21-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β ((VtxDegβπ)βπ) β β0) | ||
Theorem | eupth2lem3lem2 29746 | Lemma for eupth2lem3 29753. (Contributed by AV, 21-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β ((VtxDegβπ)βπ) β β0) | ||
Theorem | eupth2lem3lem3 29747* | Lemma for eupth2lem3 29753, formerly part of proof of eupth2lem3 29753: If a loop {(πβπ), (πβ(π + 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) & β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπ)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) & β’ (π β if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ)))) β β’ ((π β§ (πβπ) = (πβ(π + 1))) β (Β¬ 2 β₯ (((VtxDegβπ)βπ) + ((VtxDegβπ)βπ)) β π β if((πβ0) = (πβ(π + 1)), β , {(πβ0), (πβ(π + 1))}))) | ||
Theorem | eupth2lem3lem4 29748* | Lemma for eupth2lem3 29753, formerly part of proof of eupth2lem3 29753: 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) & β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπ)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) & β’ (π β if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ)))) & β’ (π β (πΌβ(πΉβπ)) β π« π) β β’ ((π β§ (πβπ) β (πβ(π + 1)) β§ (π = (πβπ) β¨ π = (πβ(π + 1)))) β (Β¬ 2 β₯ (((VtxDegβπ)βπ) + ((VtxDegβπ)βπ)) β π β if((πβ0) = (πβ(π + 1)), β , {(πβ0), (πβ(π + 1))}))) | ||
Theorem | eupth2lem3lem5 29749* | Lemma for eupth2 29756. (Contributed by AV, 25-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) & β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπ)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) & β’ (π β (πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}) β β’ (π β (πΌβ(πΉβπ)) β π« π) | ||
Theorem | eupth2lem3lem6 29750* | Formerly part of proof of eupth2lem3 29753: 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 (πβ0) and (πβπ): 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) & β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπ)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) & β’ (π β (πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}) β β’ ((π β§ (πβπ) β (πβ(π + 1)) β§ (π β (πβπ) β§ π β (πβ(π + 1)))) β (Β¬ 2 β₯ (((VtxDegβπ)βπ) + ((VtxDegβπ)βπ)) β π β if((πβ0) = (πβ(π + 1)), β , {(πβ0), (πβ(π + 1))}))) | ||
Theorem | eupth2lem3lem7 29751* | Lemma for eupth2lem3 29753: Combining trlsegvdeg 29744, eupth2lem3lem3 29747, eupth2lem3lem4 29748 and eupth2lem3lem6 29750. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 27-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) & β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπ)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) & β’ (π β (πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}) β β’ (π β (Β¬ 2 β₯ ((VtxDegβπ)βπ) β π β if((πβ0) = (πβ(π + 1)), β , {(πβ0), (πβ(π + 1))}))) | ||
Theorem | eupthvdres 29752 | Formerly part of proof of eupth2 29756: 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΊ β π) & β’ (π β Fun πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ π» = β¨π, (πΌ βΎ (πΉ β (0..^(β―βπΉ))))β© β β’ (π β (VtxDegβπ») = (VtxDegβπΊ)) | ||
Theorem | eupth2lem3 29753* | Lemma for eupth2 29756. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΊ β UPGraph) & β’ (π β Fun πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ π» = β¨π, (πΌ βΎ (πΉ β (0..^π)))β© & β’ π = β¨π, (πΌ βΎ (πΉ β (0..^(π + 1))))β© & β’ (π β π β β0) & β’ (π β (π + 1) β€ (β―βπΉ)) & β’ (π β π β π) & β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπ»)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) β β’ (π β (Β¬ 2 β₯ ((VtxDegβπ)βπ) β π β if((πβ0) = (πβ(π + 1)), β , {(πβ0), (πβ(π + 1))}))) | ||
Theorem | eupth2lemb 29754* | Lemma for eupth2 29756 (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 29756. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΊ β UPGraph) & β’ (π β Fun πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) β β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegββ¨π, (πΌ βΎ (πΉ β (0..^0)))β©)βπ₯)} = β ) | ||
Theorem | eupth2lems 29755* | Lemma for eupth2 29756 (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 29756. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΊ β UPGraph) & β’ (π β Fun πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) β β’ ((π β§ π β β0) β ((π β€ (β―βπΉ) β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegββ¨π, (πΌ βΎ (πΉ β (0..^π)))β©)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) β ((π + 1) β€ (β―βπΉ) β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegββ¨π, (πΌ βΎ (πΉ β (0..^(π + 1))))β©)βπ₯)} = if((πβ0) = (πβ(π + 1)), β , {(πβ0), (πβ(π + 1))})))) | ||
Theorem | eupth2 29756* | 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΊ β UPGraph) & β’ (π β Fun πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) β β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)} = if((πβ0) = (πβ(β―βπΉ)), β , {(πβ0), (πβ(β―βπΉ))})) | ||
Theorem | eulerpathpr 29757* | 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.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β (β―β{π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)}) β {0, 2}) | ||
Theorem | eulerpath 29758* | 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.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β UPGraph β§ (EulerPathsβπΊ) β β ) β (β―β{π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)}) β {0, 2}) | ||
Theorem | eulercrct 29759* | A pseudograph with an Eulerian circuit β¨πΉ, πβ© (an "Eulerian pseudograph") has only vertices of even degree. (Contributed by AV, 12-Mar-2021.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π β§ πΉ(CircuitsβπΊ)π) β βπ₯ β π 2 β₯ ((VtxDegβπΊ)βπ₯)) | ||
Theorem | eucrctshift 29760* | Cyclically shifting the indices of an Eulerian circuit β¨πΉ, πβ© results in an Eulerian circuit β¨π», πβ©. (Contributed by AV, 15-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΉ(CircuitsβπΊ)π) & β’ π = (β―βπΉ) & β’ (π β π β (0..^π)) & β’ π» = (πΉ cyclShift π) & β’ π = (π₯ β (0...π) β¦ if(π₯ β€ (π β π), (πβ(π₯ + π)), (πβ((π₯ + π) β π)))) & β’ (π β πΉ(EulerPathsβπΊ)π) β β’ (π β (π»(EulerPathsβπΊ)π β§ π»(CircuitsβπΊ)π)) | ||
Theorem | eucrct2eupth1 29761 | Removing one edge (πΌβ(πΉβπ)) from a nonempty graph πΊ with an Eulerian circuit β¨πΉ, πβ© results in a graph π with an Eulerian path β¨π», πβ©. This is the special case of eucrct2eupth 29762 (with π½ = (π β 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ (π β πΉ(CircuitsβπΊ)π) & β’ (Vtxβπ) = π & β’ (π β 0 < (β―βπΉ)) & β’ (π β π = ((β―βπΉ) β 1)) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ π» = (πΉ prefix π) & β’ π = (π βΎ (0...π)) β β’ (π β π»(EulerPathsβπ)π) | ||
Theorem | eucrct2eupth 29762* | Removing one edge (πΌβ(πΉβπ½)) from a graph πΊ with an Eulerian circuit β¨πΉ, πβ© results in a graph π with an Eulerian path β¨π», πβ©. (Contributed by AV, 17-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ (π β πΉ(CircuitsβπΊ)π) & β’ (Vtxβπ) = π & β’ (π β π = (β―βπΉ)) & β’ (π β π½ β (0..^π)) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β ((0..^π) β {π½})))) & β’ πΎ = (π½ + 1) & β’ π» = ((πΉ cyclShift πΎ) prefix (π β 1)) & β’ π = (π₯ β (0..^π) β¦ if(π₯ β€ (π β πΎ), (πβ(π₯ + πΎ)), (πβ((π₯ + πΎ) β π)))) β β’ (π β π»(EulerPathsβπ)π) | ||
According to Wikipedia ("Seven Bridges of KΓΆnigsberg", 9-Mar-2021, https://en.wikipedia.org/wiki/Seven_Bridges_of_Koenigsberg): "The Seven Bridges of KΓΆnigsberg 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 KΓΆnigsberg 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 KΓΆnigsberg 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 KΓΆnigsberg graph πΊ is a multigraph consisting of 4 vertices and 7 edges, represented by the following ordered pair: πΊ = β¨(0...3), β¨β{0, 1}{0, 2} {0, 3}{1, 2}{1, 2}{2, 3}{2, 3}ββ©β©, see konigsbergumgr 29768. konigsberg 29774 shows that the KΓΆnigsberg graph has no Eulerian path, thus the KΓΆnigsberg Bridge problem has no solution. | ||
Theorem | konigsbergvtx 29763 | The set of vertices of the KΓΆnigsberg graph πΊ. (Contributed by AV, 28-Feb-2021.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ (VtxβπΊ) = (0...3) | ||
Theorem | konigsbergiedg 29764 | The indexed edges of the KΓΆnigsberg graph πΊ. (Contributed by AV, 28-Feb-2021.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ (iEdgβπΊ) = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© | ||
Theorem | konigsbergiedgw 29765* | The indexed edges of the KΓΆnigsberg graph πΊ is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ πΈ β Word {π₯ β π« π β£ (β―βπ₯) = 2} | ||
Theorem | konigsbergssiedgwpr 29766* | Each subset of the indexed edges of the KΓΆnigsberg graph πΊ is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ ((π΄ β Word V β§ π΅ β Word V β§ πΈ = (π΄ ++ π΅)) β π΄ β Word {π₯ β π« π β£ (β―βπ₯) = 2}) | ||
Theorem | konigsbergssiedgw 29767* | Each subset of the indexed edges of the KΓΆnigsberg graph πΊ is a word over the pairs of vertices. (Contributed by AV, 28-Feb-2021.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ ((π΄ β Word V β§ π΅ β Word V β§ πΈ = (π΄ ++ π΅)) β π΄ β Word {π₯ β (π« π β {β }) β£ (β―βπ₯) β€ 2}) | ||
Theorem | konigsbergumgr 29768 | The KΓΆnigsberg graph πΊ is a multigraph. (Contributed by AV, 28-Feb-2021.) (Revised by AV, 9-Mar-2021.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ πΊ β UMGraph | ||
Theorem | konigsberglem1 29769 | Lemma 1 for konigsberg 29774: 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.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ ((VtxDegβπΊ)β0) = 3 | ||
Theorem | konigsberglem2 29770 | Lemma 2 for konigsberg 29774: 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.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ ((VtxDegβπΊ)β1) = 3 | ||
Theorem | konigsberglem3 29771 | Lemma 3 for konigsberg 29774: 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.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ ((VtxDegβπΊ)β3) = 3 | ||
Theorem | konigsberglem4 29772* | Lemma 4 for konigsberg 29774: Vertices 0, 1, 3 are vertices of odd degree. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ {0, 1, 3} β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)} | ||
Theorem | konigsberglem5 29773* | Lemma 5 for konigsberg 29774: The set of vertices of odd degree is greater than 2. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 28-Feb-2021.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ 2 < (β―β{π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)}) | ||
Theorem | konigsberg 29774 | The KΓΆnigsberg Bridge problem. If πΊ is the KΓΆnigsberg 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 29758 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.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ (EulerPathsβπΊ) = β | ||
Syntax | cfrgr 29775 | Extend class notation with friendship graphs. |
class FriendGraph | ||
Definition | df-frgr 29776* | 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.) |
β’ FriendGraph = {π β USGraph β£ [(Vtxβπ) / π£][(Edgβπ) / π]βπ β π£ βπ β (π£ β {π})β!π₯ β π£ {{π₯, π}, {π₯, π}} β π} | ||
Theorem | isfrgr 29777* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β FriendGraph β (πΊ β USGraph β§ βπ β π βπ β (π β {π})β!π₯ β π {{π₯, π}, {π₯, π}} β πΈ)) | ||
Theorem | frgrusgr 29778 | A friendship graph is a simple graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) |
β’ (πΊ β FriendGraph β πΊ β USGraph) | ||
Theorem | frgr0v 29779 | 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.) |
β’ ((πΊ β π β§ (VtxβπΊ) = β ) β (πΊ β FriendGraph β (iEdgβπΊ) = β )) | ||
Theorem | frgr0vb 29780 | 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.) |
β’ ((πΊ β π β§ (VtxβπΊ) = β β§ (iEdgβπΊ) = β ) β πΊ β FriendGraph ) | ||
Theorem | frgruhgr0v 29781 | Any null graph (without vertices) represented as hypergraph is a friendship graph. (Contributed by AV, 29-Mar-2021.) |
β’ ((πΊ β UHGraph β§ (VtxβπΊ) = β ) β πΊ β FriendGraph ) | ||
Theorem | frgr0 29782 | The null graph (graph without vertices) is a friendship graph. (Contributed by AV, 29-Mar-2021.) |
β’ β β FriendGraph | ||
Theorem | frcond1 29783* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β FriendGraph β ((π΄ β π β§ πΆ β π β§ π΄ β πΆ) β β!π β π {{π΄, π}, {π, πΆ}} β πΈ)) | ||
Theorem | frcond2 29784* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β FriendGraph β ((π΄ β π β§ πΆ β π β§ π΄ β πΆ) β β!π β π ({π΄, π} β πΈ β§ {π, πΆ} β πΈ))) | ||
Theorem | frgreu 29785* | Variant of frcond2 29784: 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β FriendGraph β ((π΄ β π β§ πΆ β π β§ π΄ β πΆ) β β!π({π΄, π} β πΈ β§ {π, πΆ} β πΈ))) | ||
Theorem | frcond3 29786* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β FriendGraph β ((π΄ β π β§ πΆ β π β§ π΄ β πΆ) β βπ₯ β π ((πΊ NeighbVtx π΄) β© (πΊ NeighbVtx πΆ)) = {π₯})) | ||
Theorem | frcond4 29787* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β FriendGraph β βπ β π βπ β (π β {π})βπ₯ β π ((πΊ NeighbVtx π) β© (πΊ NeighbVtx π)) = {π₯}) | ||
Theorem | frgr1v 29788 | 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.) |
β’ ((πΊ β USGraph β§ (VtxβπΊ) = {π}) β πΊ β FriendGraph ) | ||
Theorem | nfrgr2v 29789 | 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.) |
β’ (((π΄ β π β§ π΅ β π β§ π΄ β π΅) β§ (VtxβπΊ) = {π΄, π΅}) β πΊ β FriendGraph ) | ||
Theorem | frgr3vlem1 29790* | Lemma 1 for frgr3v 29792. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (((π΄ β π β§ π΅ β π β§ πΆ β π) β§ (π΄ β π΅ β§ π΄ β πΆ β§ π΅ β πΆ) β§ (π = {π΄, π΅, πΆ} β§ πΊ β USGraph)) β βπ₯βπ¦(((π₯ β {π΄, π΅, πΆ} β§ {{π₯, π΄}, {π₯, π΅}} β πΈ) β§ (π¦ β {π΄, π΅, πΆ} β§ {{π¦, π΄}, {π¦, π΅}} β πΈ)) β π₯ = π¦)) | ||
Theorem | frgr3vlem2 29791* | Lemma 2 for frgr3v 29792. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (((π΄ β π β§ π΅ β π β§ πΆ β π) β§ (π΄ β π΅ β§ π΄ β πΆ β§ π΅ β πΆ)) β ((π = {π΄, π΅, πΆ} β§ πΊ β USGraph) β (β!π₯ β {π΄, π΅, πΆ} {{π₯, π΄}, {π₯, π΅}} β πΈ β ({πΆ, π΄} β πΈ β§ {πΆ, π΅} β πΈ)))) | ||
Theorem | frgr3v 29792 | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (((π΄ β π β§ π΅ β π β§ πΆ β π) β§ (π΄ β π΅ β§ π΄ β πΆ β§ π΅ β πΆ)) β ((π = {π΄, π΅, πΆ} β§ πΊ β USGraph) β (πΊ β FriendGraph β ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)))) | ||
Theorem | 1vwmgr 29793* | 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.) |
β’ ((π΄ β π β§ π = {π΄}) β ββ β π βπ£ β (π β {β})({π£, β} β πΈ β§ β!π€ β (π β {β}){π£, π€} β πΈ)) | ||
Theorem | 3vfriswmgrlem 29794* | Lemma for 3vfriswmgr 29795. (Contributed by Alexander van der Vekens, 6-Oct-2017.) (Revised by AV, 31-Mar-2021.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (((π΄ β π β§ π΅ β π β§ π΄ β π΅) β§ (π = {π΄, π΅, πΆ} β§ πΊ β USGraph)) β ({π΄, π΅} β πΈ β β!π€ β {π΄, π΅} {π΄, π€} β πΈ)) | ||
Theorem | 3vfriswmgr 29795* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (((π΄ β π β§ π΅ β π β§ πΆ β π) β§ (π΄ β π΅ β§ π΄ β πΆ β§ π΅ β πΆ) β§ π = {π΄, π΅, πΆ}) β (πΊ β FriendGraph β ββ β π βπ£ β (π β {β})({π£, β} β πΈ β§ β!π€ β (π β {β}){π£, π€} β πΈ))) | ||
Theorem | 1to2vfriswmgr 29796* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ ((π΄ β π β§ (π = {π΄} β¨ π = {π΄, π΅})) β (πΊ β FriendGraph β ββ β π βπ£ β (π β {β})({π£, β} β πΈ β§ β!π€ β (π β {β}){π£, π€} β πΈ))) | ||
Theorem | 1to3vfriswmgr 29797* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ ((π΄ β π β§ (π = {π΄} β¨ π = {π΄, π΅} β¨ π = {π΄, π΅, πΆ})) β (πΊ β FriendGraph β ββ β π βπ£ β (π β {β})({π£, β} β πΈ β§ β!π€ β (π β {β}){π£, π€} β πΈ))) | ||
Theorem | 1to3vfriendship 29798* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ ((π΄ β π β§ (π = {π΄} β¨ π = {π΄, π΅} β¨ π = {π΄, π΅, πΆ})) β (πΊ β FriendGraph β βπ£ β π βπ€ β (π β {π£}){π£, π€} β πΈ)) | ||
Theorem | 2pthfrgrrn 29799* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β FriendGraph β βπ β π βπ β (π β {π})βπ β π ({π, π} β πΈ β§ {π, π} β πΈ)) | ||
Theorem | 2pthfrgrrn2 29800* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β FriendGraph β βπ β π βπ β (π β {π})βπ β π (({π, π} β πΈ β§ {π, π} β πΈ) β§ (π β π β§ π β π))) |
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