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Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremuvtx01vtx 21501* If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. This theorem could have been stated UnivVertex , but a lot of auxiliary theorems would have been needed. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
UnivVertex

Theoremuvtxnbgra 21502 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
USGrph UnivVertex Neighbors

Theoremuvtxnm1nbgra 21503 A universal vertex has neighbors in a graph with vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
USGrph UnivVertex Neighbors

Theoremuvtxnbgravtx 21504* A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
USGrph UnivVertex Neighbors

Theoremcusgrauvtxb 21505 An undirected simple graph is complete if and only if each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.)
USGrph ComplUSGrph UnivVertex

14.1.5  Walks, paths and cycles

Syntaxcwalk 21506 Extend class notation with Walks (of a graph).
Walks

Syntaxctrail 21507 Extend class notation with Trails (of a graph).
Trails

Syntaxcpath 21508 Extend class notation with Paths (of a graph).
Paths

Syntaxcspath 21509 Extend class notation with Simple Paths (of a graph).
SPaths

Syntaxcwlkon 21510 Extend class notation with Walks between two vertices (within a graph).
WalkOn

Syntaxctrlon 21511 Extend class notation with Trails between two vertices (within a graph).
TrailOn

Syntaxcpthon 21512 Extend class notation with Paths between two vertices (within a graph).
PathOn

Syntaxcspthon 21513 Extend class notation with simple paths between two vertices (within a graph).
SPathOn

Syntaxccrct 21514 Extend class notation with Circuits (of a graph).
Circuits

Syntaxccycl 21515 Extend class notation with Cycles (of a graph).
Cycles

Definitiondf-wlk 21516* Define the set of all Walks (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)."

According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4.

Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Walks Word ..^

Definitiondf-trail 21517* Define the set of all Trails (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A trail is a walk in which all edges are distinct.

According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5.

Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Trails Walks

Definitiondf-pth 21518* Define the set of all Paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

According to Bollobas: "... a path is a walk with distinct vertices.", see Notation of [Bollobas] p. 5. (A walk with distinct vertices is actually a simple path, see wlkdvspth 21608).

Therefore, a path can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, which is injective restricted to the set { 1 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Paths Trails ..^ ..^

Definitiondf-spth 21519* Define the set of all Simple Paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) (Contributed by Alexander van der Vekens, 20-Oct-2017.)

SPaths Trails

Definitiondf-crct 21520* Define the set of all circuits (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A circuit can be a closed walk allowing repetitions of vertices but not edges;"; according to Wikipedia ("Glossary of graph theory terms", https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms, 3-Oct-2017): "A circuit may refer to ... a trail (a closed tour without repeated edges), ...".

Following Bollobas ("A trail whose endvertices coincide (a closed trail) is called a circuit.", see Definition of [Bollobas] p. 5.), a circuit is a closed trail without repeated edges. So the circuit is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) (Contributed by Alexander van der Vekens, 3-Oct-2017.)

Circuits Trails

Definitiondf-cycl 21521* Define the set of all (simple) cycles (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex,"

According to Bollobas: "If a walk W = x0 x1 ... x(l) is such that l >= 3, x0=x(l), and the vertices x(i), 0 < i < l, are distinct from each other and x0, then W is said to be a cycle.", see Definition of [Bollobas] p. 5.

However, since a walk consisting of distinct vertices (except the first and the last vertex) is a path, a cycle can be defined as path whose first and last vertices coincide. So a cycle is represented by the following sequence: p(0) e(f(1)) p(1) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.)

Cycles Paths

Definitiondf-wlkon 21522* Define the collection of walks with particular endpoints (in an un- directed graph). This corresponds to the "x0-x(l)-walks", see Definition in [Bollobas] p. 5. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
WalkOn Walks

Definitiondf-trlon 21523* Define the collection of trails with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
TrailOn WalkOn Trails

Definitiondf-pthon 21524* Define the collection of paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
PathOn WalkOn Paths

Definitiondf-spthon 21525* Define the collection of simple paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
SPathOn WalkOn SPaths

14.1.5.1  Walks and trails

Theoremwlks 21526* The set of walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.)
Walks Word ..^

Theoremiswlk 21527* Properties of a pair of functions to be a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Walks Word ..^

Theorem2mwlk 21528 The two mappings determining a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Walks Word

Theoremwlkres 21529* Restrictions of walks are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
Walks

Theoremwlkon 21530* The set of walks between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 12-Dec-2017.)
WalkOn Walks

Theoremiswlkon 21531 Properties of a pair of functions to be a walk between two given vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
WalkOn Walks

Theoremwlkonprop 21532 Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
WalkOn Walks

Theoremwlkoniswlk 21533 A walk between to vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
WalkOn Walks

Theoremwlkbprop 21534 Basic properties of a walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
Walks

Theoremwlkonwlk 21535 A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.)
Walks WalkOn

Theoremtrls 21536* The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Trails Word ..^

Theoremistrl 21537* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Trails Word ..^

Theoremistrl2 21538* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Trails ..^ ..^

Theoremtrliswlk 21539 A trail is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Trails Walks

Theoremtrlon 21540* The set of trails between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 4-Nov-2017.)
TrailOn WalkOn Trails

Theoremistrlon 21541 Properties of a pair of functions to be a trail between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
TrailOn WalkOn Trails

Theoremtrlonprop 21542 Properties of a trail between two vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
TrailOn WalkOn Trails

Theoremtrlonistrl 21543 A trail between to vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
TrailOn Trails

Theoremtrlonwlkon 21544 A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
TrailOn WalkOn

Theorem0wlk 21545 A pair of an empty set (of edges) and a second set (of vertices) is a walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Walks

Theorem0trl 21546 A pair of an empty set (of edges) and a second set (of vertices) is a trail if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Trails

Theorem0wlkon 21547 A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
WalkOn

Theorem0trlon 21548 A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
TrailOn

Theorem2trllemF 21549 Lemma 5 for constr2trl 21599. (Contributed by Alexander van der Vekens, 31-Jan-2018.)

Theorem2trllemA 21550 Lemma 1 for constr2trl 21599. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)

Theorem2trllemB 21551 Lemma 2 for constr2trl 21599. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
..^

Theorem2trllemH 21552 Lemma 3 for constr2trl 21599. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
..^

Theorem2trllemE 21553 Lemma 4 for constr2trl 21599. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
..^

Theorem2wlklemA 21554 Lemma for constr2wlk 21598. (Contributed by Alexander van der Vekens, 18-Feb-2018.)

Theorem2wlklemB 21555 Lemma for constr2wlk 21598. (Contributed by Alexander van der Vekens, 18-Feb-2018.)

Theorem2wlklemC 21556 Lemma for constr2wlk 21598. (Contributed by Alexander van der Vekens, 18-Feb-2018.)

Theorem2trllemD 21557 Lemma 4 for constr2trl 21599. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)

Theorem2trllemG 21558 Lemma 7 for constr2trl 21599. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Theoremwlkntrllem1 21559 Lemma 1 for wlkntrl 21562: F is a word over , the domain of E. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
Word

Theoremwlkntrllem2 21560* Lemma 2 for wlkntrl 21562: The values of E after F are edges between two vertices enumerated by P. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
..^

Theoremwlkntrllem3 21561* Lemma 3 for wlkntrl 21562: F is not injective. (Contributed by Alexander van der Vekens, 22-Oct-2017.)

Theoremwlkntrl 21562* A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one edge to the other is a walk, but not a trail. Notice that is a simple graph (without loops) only if . (Contributed by Alexander van der Vekens, 22-Oct-2017.)
Walks Trails

Theoremusgrnloop 21563* In an undirected simple graph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.)
USGrph Walks ..^

Theorem2wlklem 21564* Lemma for is2wlk 21565 and 2wlklemA 21554. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Theoremis2wlk 21565 Properties of a pair of functions to be a walk of length 2 (in an undirected graph). (Contributed by Alexander van der Vekens, 16-Feb-2018.)
Walks ..^

14.1.5.2  Paths and simple paths

Theorempths 21566* The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Paths Trails ..^ ..^

Theoremspths 21567* The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
SPaths Trails

Theoremispth 21568 Properties of a pair of functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
Paths Trails ..^ ..^

Theoremisspth 21569 Properties of a pair of functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
SPaths Trails

Theorem0pth 21570 A pair of an empty set (of edges) and a second set (of vertices) is a path if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Paths

Theorem0spth 21571 A pair of an empty set (of edges) and a second set (of vertices) is a simple path if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
SPaths

Theorempthistrl 21572 A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
Paths Trails

Theoremspthispth 21573 A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
SPaths Paths

Theorempthdepisspth 21574 A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.)
Paths SPaths

Theorempthon 21575* The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)
PathOn WalkOn Paths

Theoremispthon 21576 Properties of a pair of functions to be a path between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)
PathOn WalkOn Paths

Theorempthonprop 21577 Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
PathOn WalkOn Paths

Theorempthonispth 21578 A path between two vertices is a path. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
PathOn Paths

Theorem0pthon 21579 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
PathOn

Theorem0pthon1 21580 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
PathOn

Theorem0pthonv 21581* For each vertex there is a path of length 0 from the vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
PathOn

Theoremspthon 21582* The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
SPathOn WalkOn SPaths

Theoremisspthon 21583 Properties of a pair of functions to be a simple path between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
SPathOn WalkOn SPaths

Theoremisspthonpth 21584 Properties of a pair of functions to be a simple path between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 9-Mar-2018.)
SPathOn SPaths

Theoremspthonprp 21585 Properties of a simple path between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
SPathOn WalkOn SPaths

Theoremspthonisspth 21586 A simple path between to vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
SPathOn SPaths

Theoremspthonepeq 21587 The endpoints of a simple path between two vertices are equal if and only if the path is of length 0 (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
SPathOn

Theoremconstr1trl 21588 Construction of a trail from one given edge in a graph. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
Trails

Theorem1pthonlem1 21589 Lemma 1 for 1pthon 21591. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
..^

Theorem1pthonlem2 21590 Lemma 2 for 1pthon 21591. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
..^

Theorem1pthon 21591 A path of length 1 from one vertex to another vertex. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
PathOn

Theorem1pthoncl 21592 A path of length 1 from one vertex to another vertex. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
PathOn

Theorem1pthon2v 21593* For each pair of adjacent vertices there is a path of length 1 from one vertex to the other. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
PathOn

Theoremconstr2spthlem1 21594 Lemma 1 for constr2spth 21600. (Contributed by Alexander van der Vekens, 31-Jan-2018.)

Theorem2pthlem1 21595 Lemma 1 for constr2pth 21601. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
..^

Theorem2pthlem2 21596 Lemma 2 for constr2pth 21601. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Revised by Alexander van der Vekens, 18-Feb-2018.)
..^

Theorem2wlklem1 21597* Lemma 1 for constr2wlk 21598. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Theoremconstr2wlk 21598 Construction of a walk from two given edges in a graph. (Contributed by Alexander van der Vekens, 5-Feb-2018.)
Walks

Theoremconstr2trl 21599 Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 1-Feb-2018.)
Trails

Theoremconstr2spth 21600 A simple path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
SPaths

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