![]() |
Metamath
Proof Explorer Theorem List (p. 297 of 491) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30946) |
![]() (30947-32469) |
![]() (32470-49035) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fusgrregdegfi 29601* | In a nonempty finite simple graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 6-Mar-2018.) (Revised by AV, 19-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → 𝐾 ∈ ℕ0)) | ||
Theorem | fusgrn0eqdrusgr 29602* | If all vertices in a nonempty finite simple graph have the same (finite) degree, the graph is k-regular. (Contributed by AV, 26-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → 𝐺 RegUSGraph 𝐾)) | ||
Theorem | frusgrnn0 29603 | In a nonempty finite k-regular simple graph, the degree of each vertex is finite. (Contributed by AV, 7-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0) | ||
Theorem | 0edg0rgr 29604 | A graph is 0-regular if it has no edges. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 RegGraph 0) | ||
Theorem | uhgr0edg0rgr 29605 | A hypergraph is 0-regular if it has no edges. (Contributed by AV, 19-Dec-2020.) |
⊢ ((𝐺 ∈ UHGraph ∧ (Edg‘𝐺) = ∅) → 𝐺 RegGraph 0) | ||
Theorem | uhgr0edg0rgrb 29606 | A hypergraph is 0-regular iff it has no edges. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 24-Dec-2020.) |
⊢ (𝐺 ∈ UHGraph → (𝐺 RegGraph 0 ↔ (Edg‘𝐺) = ∅)) | ||
Theorem | usgr0edg0rusgr 29607 | A simple graph is 0-regular iff it has no edges. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 19-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.) |
⊢ (𝐺 ∈ USGraph → (𝐺 RegUSGraph 0 ↔ (Edg‘𝐺) = ∅)) | ||
Theorem | 0vtxrgr 29608* | A null graph (with no vertices) is k-regular for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegGraph 𝑘) | ||
Theorem | 0vtxrusgr 29609* | A graph with no vertices and an empty edge function is a k-regular simple graph for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘) | ||
Theorem | 0uhgrrusgr 29610* | The null graph as hypergraph is a k-regular simple graph for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘) | ||
Theorem | 0grrusgr 29611 | The null graph represented by an empty set is a k-regular simple graph for every k. (Contributed by AV, 26-Dec-2020.) |
⊢ ∀𝑘 ∈ ℕ0* ∅ RegUSGraph 𝑘 | ||
Theorem | 0grrgr 29612 | The null graph represented by an empty set is k-regular for every k. (Contributed by AV, 26-Dec-2020.) |
⊢ ∀𝑘 ∈ ℕ0* ∅ RegGraph 𝑘 | ||
Theorem | cusgrrusgr 29613 | A complete simple graph with n vertices (at least one) is (n-1)-regular. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝐺 RegUSGraph ((♯‘𝑉) − 1)) | ||
Theorem | cusgrm1rusgr 29614 | A finite simple graph with n vertices is complete iff it is (n-1)-regular. Hint: If the definition of RegGraph was allowed for 𝑘 ∈ ℤ, then the assumption 𝑉 ≠ ∅ could be removed. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (𝐺 ∈ ComplUSGraph ↔ 𝐺 RegUSGraph ((♯‘𝑉) − 1))) | ||
Theorem | rusgrpropnb 29615* | The properties of a k-regular simple graph expressed with neighbors. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾)) | ||
Theorem | rusgrpropedg 29616* | The properties of a k-regular simple graph expressed with edges. (Contributed by AV, 23-Dec-2020.) (Revised by AV, 27-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾)) | ||
Theorem | rusgrpropadjvtx 29617* | The properties of a k-regular simple graph expressed with adjacent vertices. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 27-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾)) | ||
Theorem | rusgrnumwrdl2 29618* | In a k-regular simple graph, the number of edges resp. walks of length 1 (represented as words of length 2) starting at a fixed vertex is k. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 6-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = 𝐾) | ||
Theorem | rusgr1vtxlem 29619* | Lemma for rusgr1vtx 29620. (Contributed by AV, 27-Dec-2020.) |
⊢ (((∀𝑣 ∈ 𝑉 (♯‘𝐴) = 𝐾 ∧ ∀𝑣 ∈ 𝑉 𝐴 = ∅) ∧ (𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 1)) → 𝐾 = 0) | ||
Theorem | rusgr1vtx 29620 | If a k-regular simple graph has only one vertex, then k must be 0. (Contributed by Alexander van der Vekens, 4-Sep-2018.) (Revised by AV, 27-Dec-2020.) |
⊢ (((♯‘(Vtx‘𝐺)) = 1 ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 = 0) | ||
Theorem | rgrusgrprc 29621* | The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.) |
⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V | ||
Theorem | rusgrprc 29622 | The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.) |
⊢ {𝑔 ∣ 𝑔 RegUSGraph 0} ∉ V | ||
Theorem | rgrprc 29623 | The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.) |
⊢ {𝑔 ∣ 𝑔 RegGraph 0} ∉ V | ||
Theorem | rgrprcx 29624* | The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.) |
⊢ {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V | ||
Theorem | rgrx0ndm 29625* | 0 is not in the domain of the potentially alternative definition of the sets of k-regular graphs for each extended nonnegative integer k. (Contributed by AV, 28-Dec-2020.) |
⊢ 𝑅 = (𝑘 ∈ ℕ0* ↦ {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘}) ⇒ ⊢ 0 ∉ dom 𝑅 | ||
Theorem | rgrx0nd 29626* | The potentially alternatively defined k-regular graphs is not defined for k=0. (Contributed by AV, 28-Dec-2020.) |
⊢ 𝑅 = (𝑘 ∈ ℕ0* ↦ {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘}) ⇒ ⊢ (𝑅‘0) = ∅ | ||
A "walk" in a graph is usually defined for simple graphs, multigraphs or even pseudographs as "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, or "A walk (of length k) in a graph is a nonempty alternating sequence v0 e0 v1 e1 ... e(k-1) vk of vertices and edges in G such that ei = { vi , vi+1 } for all i < k.", see definition of [Diestel] p. 10. Formalizing these definitions (mainly by representing the indexed vertices and edges by functions), a walk is represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges (e is a third function enumerating the edges within the graph, not within the walk), and p enumerates the vertices, see df-wlks 29631. Hence a walk (of length n) is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Alternatively, one could define a walk as a function 𝑤:(0...(2 · 𝑛))⟶((Edg‘𝐺) ∪ (Vtx‘𝐺)) such that for all 0 ≤ 𝑘 ≤ 𝑛, (𝑤‘(2 · 𝑘)) ∈ (Vtx‘𝐺) and for all 0 ≤ 𝑘 ≤ (𝑛 − 1), (𝑤‘((2 · 𝑘) + 1)) ∈ (Edg‘𝐺) and {(𝑤‘(2 · 𝑘)), (𝑤‘((2 · 𝑘) + 2))} ⊆ (𝑤‘((2 · 𝑘) + 1)). Based on our definition of Walks, the class of all walks, more restrictive constructs are defined: * Trails (df-trls 29724): A "walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5, i.e., f(i) =/= f(j) if i =/= j. * Paths (df-pths 29748): A path is a walk whose vertices except the first and the last vertex are distinct, i.e., p(i) =/= p(j) if i < j, except possibly when i = 0 and j = n. * SPaths (simple paths, df-spths 29749): A simple path "is a walk with distinct vertices.", see Notation of [Bollobas] p. 5, i.e., p(i) =/= p(j) if i =/= j. * ClWalks (closed walks, df-clwlks 29803): A walk whose endvertices coincide is called a closed walk, i.e., p(0) = p(n). * Circuits (df-crcts 29818): "A trail whose endvertices coincide (a closed trail) is called a circuit." (see Definition of [Bollobas] p. 5), i.e., f(i) =/= f(j) if i =/= j and p(0) = p(n). Equivalently, a circuit is a closed walk with distinct edges. * Cycles (df-cycls 29819): A path whose endvertices coincide (a closed path) is called a cycle, i.e., p(i) =/= p(j) if i =/= j, except i = 0 and j = n, and p(0) = p(n). Equivalently, a cycle is a closed walk with distinct vertices. * EulerPaths (Eulerian paths, df-eupth 30226): An Eulerian path is "a trail containing all edges [of the graph]" (see definition in [Bollobas] p. 16), i.e., f(i) =/= f(j) if i =/= j and for all edges e(x) there is an 1 <= i <= n with e(x) = e(f(i)). Note, however, that an Eulerian path needs not be a path. * Eulerian circuit: 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), i.e., f(i) =/= f(j) if i =/= j, p(0) = p(n) and for all edges e(x) there is an 1 <= i <= n with e(x) = e(f(i)). Hierarchy of all kinds of walks (apply ssriv 3998 and elopabran 5571 to the mentioned theorems to obtain the following subset relationships, as available for clwlkiswlk 29806, see clwlkwlk 29807 and clwlkswks 29808): * Trails are walks (trliswlk 29729): (Trails‘𝐺) ⊆ (Walks‘𝐺) * Paths are trails (pthistrl 29757): (Paths‘𝐺) ⊆ (Trails‘𝐺) * Simple paths are paths (spthispth 29758): (SPaths‘𝐺) ⊆ (Paths‘𝐺) * Closed walks are walks (clwlkiswlk 29806): (ClWalks‘𝐺) ⊆ (Walks‘𝐺) * Circuits are closed walks (crctisclwlk 29826): (Circuits‘𝐺) ⊆ (ClWalks‘𝐺) * Circuits are trails (crctistrl 29827): (Circuits‘𝐺) ⊆ (Trails‘𝐺) * Cycles are paths (cyclispth 29829): (Cycles‘𝐺) ⊆ (Paths‘𝐺) * Cycles are circuits (cycliscrct 29831): (Cycles‘𝐺) ⊆ (Circuits‘𝐺) * (Non-trivial) cycles are not simple paths (cyclnspth 29832): (𝐹 ≠ ∅ → (𝐹(Cycles‘𝐺)𝑃 → ¬ 𝐹(SPaths‘𝐺)𝑃)) * Eulerian paths are trails (eupthistrl 30239): (EulerPaths‘𝐺) ⊆ (Trails‘𝐺) Often, it is sufficient to refer to a walk by the natural sequence of its vertices, i.e., omitting its edges in its representation: p(0) p(1) ... p(n-1) p(n), see the corresponding remark in [Diestel] p. 6. The concept of a Word, see df-word 14549, is the appropriate way to define such a sequence (being finite and starting at index 0) of vertices. Therefore, it is used in definition df-wwlks 29859 for WWalks, and the representation of a walk as sequence of its vertices is called "walk as word". Only for simple pseudographs, however, the edges can be uniquely reconstructed from such a representation. In this case, the general definitions of walks and the definition of walks as words are equivalent, see wlkiswwlks 29905. In other cases, there could be more than one edge between two adjacent vertices in the walk (in a multigraph), or two adjacent vertices could be connected by two different hyperedges involving additional vertices (in a hypergraph). Based on this definition of WWalks, the class of all walks as word, more restrictive constructs are defined analogously to the general definition of a walk: * WWalksN (walks of length N as word, df-wwlksn 29860): n = N * WSPathsN (simple paths of length N as word, df-wspthsn 29862): p(i) =/= p(j) if i =/= j and n = N * ClWWalks (closed walks as word, df-clwwlk 30010): p(0) = p(n) * ClWWalksN (closed walks of length N as word, df-clwwlkn 30053): p(0) = p(n) and n = N Finally, there are a couple of definitions for (special) walks 〈𝐹, 𝑃〉 having fixed endpoints 𝐴 and 𝐵: * Walks with particular endpoints (df-wlkson 29632): 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 * Trails with particular endpoints (df-trlson 29725): 𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 * Paths with particular endpoints (df-pthson 29750): 𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 * Simple paths with particular endpoints (df-spthson 29751): 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 * Walks of a fixed length 𝑁 as words with particular endpoints (df-wwlksnon 29861): (𝐴(𝑁 WWalksNOn 𝐺)𝐵) * Simple paths of a fixed length 𝑁 as words with particular endpoints (df-wspthsnon 29863): (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) * Closed Walks of a fixed length 𝑁 as words anchored at a particular vertex 𝐴 (df-wwlksnon 29861): (𝐴(ClWWalksNOn‘𝐺)𝑁) | ||
A "walk" within a graph is usually defined for simple graphs, multigraphs or even pseudographs as "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. This definition requires the edges to connect two vertices at most (loops are also allowed: if e(i) is a loop, then x(i-1) = x(i)). For hypergraphs containing hyperedges (i.e. edges connecting more than two vertices), however, a more general definition is needed. Two approaches for a definition applicable for arbitrary hypergraphs are used in the literature: "walks on the vertex level" and "walks on the edge level" (see Aksoy, Joslyn, Marrero, Praggastis, Purvine: "Hypernetwork science via high-order hypergraph walks", June 2020, https://doi.org/10.1140/epjds/s13688-020-00231-0): "walks on the edge level": For a positive integer s, an s-walk of length k between hyperedges f and g is a sequence of hyperedges, f=e(0), e(1), ... , e(k)=g, where for j=1, ... , k, e(j-1) =/= e(j) and e(j-1) and e(j) have at least s vertices in common (according to Aksoy et al.). "walks on the vertex level": For a positive integer s, an s-walk of length k between vertices a and b is a sequence of vertices, a=v(0), v(1), ... , v(k)=b, where for j=1, ... , k, v(j-1) and v(j) are connected by at least s edges (analogous to Aksoy et al.). There are two imperfections for the definition for "walks on the edge level": one is that a walk of length 1 consists of two edges (or a walk of length 0 consists of one edge), whereas a walk of length 1 on the vertex level consists of two vertices and one edge (or a walk of length 0 consists of one vertex and no edge). The other is that edges, especially loops, can be traversed only once (and not repeatedly) because of the condition e(j-1) =/= e(j). The latter is avoided in the definition for EdgWalks, see df-ewlks 29630. To be compatible with the (usual) definition of walks for pseudographs, walks also suitable for arbitrary hypergraphs are defined "on the vertex level" in the following as Walks, see df-wlks 29631, restricting s to 1. wlk1ewlk 29672 shows that such a 1-walk "on the vertex level" induces a 1-walk "on the edge level". | ||
Syntax | cewlks 29627 | Extend class notation with s-walks "on the edge level" (of a hypergraph). |
class EdgWalks | ||
Syntax | cwlks 29628 | Extend class notation with walks (i.e. 1-walks) (of a hypergraph). |
class Walks | ||
Syntax | cwlkson 29629 | Extend class notation with walks between two vertices (within a graph). |
class WalksOn | ||
Definition | df-ewlks 29630* | Define the set of all s-walks of edges (in a hypergraph) corresponding to s-walks "on the edge level" discussed in Aksoy et al. For an extended nonnegative integer s, an s-walk is a sequence of hyperedges, e(0), e(1), ... , e(k), where e(j-1) and e(j) have at least s vertices in common (for j=1, ... , k). In contrast to the definition in Aksoy et al., 𝑠 = 0 (a 0-walk is an arbitrary sequence of hyperedges) and 𝑠 = +∞ (then the number of common vertices of two adjacent hyperedges must be infinite) are allowed. Furthermore, it is not forbidden that adjacent hyperedges are equal. (Contributed by AV, 4-Jan-2021.) |
⊢ EdgWalks = (𝑔 ∈ V, 𝑠 ∈ ℕ0* ↦ {𝑓 ∣ [(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑠 ≤ (♯‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘)))))}) | ||
Definition | df-wlks 29631* |
Define the set of all walks (in a hypergraph). Such walks correspond to
the s-walks "on the vertex level" (with s = 1), and also to
1-walks "on
the edge level" (see wlk1walk 29671) discussed in Aksoy et al. The
predicate 𝐹(Walks‘𝐺)𝑃 can be read as "The pair
〈𝐹, 𝑃〉 represents a walk in a graph
𝐺", see also iswlk 29642.
The condition {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘)) (hereinafter referred to as C) would not be sufficient, because the repetition of a vertex in a walk (i.e. (𝑝‘𝑘) = (𝑝‘(𝑘 + 1)) should be allowed only if there is a loop at (𝑝‘𝑘). Otherwise, C would be fulfilled by each edge containing (𝑝‘𝑘). According to the definition of [Bollobas] p. 4.: "A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) ...", 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 AV, 30-Dec-2020.) |
⊢ Walks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))}) | ||
Definition | df-wlkson 29632* | Define the collection of walks with particular endpoints (in a hypergraph). The predicate 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 can be read as "The pair 〈𝐹, 𝑃〉 represents a walk from vertex 𝐴 to vertex 𝐵 in a graph 𝐺", see also iswlkon 29689. 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.) (Revised by AV, 28-Dec-2020.) |
⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)})) | ||
Theorem | ewlksfval 29633* | The set of s-walks of edges (in a hypergraph). (Contributed by AV, 4-Jan-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝐺 EdgWalks 𝑆) = {𝑓 ∣ (𝑓 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘)))))}) | ||
Theorem | isewlk 29634* | Conditions for a function (sequence of hyperedges) to be an s-walk of edges. (Contributed by AV, 4-Jan-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0* ∧ 𝐹 ∈ 𝑈) → (𝐹 ∈ (𝐺 EdgWalks 𝑆) ↔ (𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))))))) | ||
Theorem | ewlkprop 29635* | Properties of an s-walk of edges. (Contributed by AV, 4-Jan-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))))) | ||
Theorem | ewlkinedg 29636 | The intersection (common vertices) of two adjacent edges in an s-walk of edges. (Contributed by AV, 4-Jan-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝐾 ∈ (1..^(♯‘𝐹))) → 𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝐾 − 1))) ∩ (𝐼‘(𝐹‘𝐾))))) | ||
Theorem | ewlkle 29637 | An s-walk of edges is also a t-walk of edges if 𝑡 ≤ 𝑠. (Contributed by AV, 4-Jan-2021.) |
⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝑇 ∈ ℕ0* ∧ 𝑇 ≤ 𝑆) → 𝐹 ∈ (𝐺 EdgWalks 𝑇)) | ||
Theorem | upgrewlkle2 29638 | In a pseudograph, there is no s-walk of edges of length greater than 1 with s>2. (Contributed by AV, 4-Jan-2021.) |
⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 1 < (♯‘𝐹)) → 𝑆 ≤ 2) | ||
Theorem | wkslem1 29639 | Lemma 1 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.) |
⊢ (𝐴 = 𝐵 → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘(𝐵 + 1)), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹‘𝐵))))) | ||
Theorem | wkslem2 29640 | Lemma 2 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.) |
⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘𝐶), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘𝐶)} ⊆ (𝐼‘(𝐹‘𝐵))))) | ||
Theorem | wksfval 29641* | The set of walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (Walks‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))))}) | ||
Theorem | iswlk 29642* | Properties of a pair of functions to be a walk. (Contributed by AV, 30-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))) | ||
Theorem | wlkprop 29643* | Properties of a walk. (Contributed by AV, 5-Nov-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))) | ||
Theorem | wlkv 29644 | The classes involved in a walk are sets. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 3-Feb-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | ||
Theorem | iswlkg 29645* | Generalization of iswlk 29642: Conditions for two classes to represent a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))) | ||
Theorem | wlkf 29646 | The mapping enumerating the (indices of the) edges of a walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) | ||
Theorem | wlkcl 29647 | A walk has length ♯(𝐹), which is an integer. Formerly proven for an Eulerian path, see eupthcl 30238. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | ||
Theorem | wlkp 29648 | The mapping enumerating the vertices of a walk is a function. (Contributed by AV, 5-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉) | ||
Theorem | wlkpwrd 29649 | The sequence of vertices of a walk is a word over the set of vertices. (Contributed by AV, 27-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word 𝑉) | ||
Theorem | wlklenvp1 29650 | The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 1-May-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1)) | ||
Theorem | wksv 29651* | The class of walks is a set. (Contributed by AV, 15-Jan-2021.) (Proof shortened by SN, 11-Dec-2024.) |
⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V | ||
Theorem | wksvOLD 29652* | Obsolete version of wksv 29651 as of 11-Dec-2024. (Contributed by AV, 15-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V | ||
Theorem | wlkn0 29653 | The sequence of vertices of a walk cannot be empty, i.e. a walk always consists of at least one vertex. (Contributed by Alexander van der Vekens, 19-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ≠ ∅) | ||
Theorem | wlklenvm1 29654 | The number of edges of a walk is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) = ((♯‘𝑃) − 1)) | ||
Theorem | ifpsnprss 29655 | Lemma for wlkvtxeledg 29656: Two adjacent (not necessarily different) vertices 𝐴 and 𝐵 in a walk are incident with an edge 𝐸. (Contributed by AV, 4-Apr-2021.) (Revised by AV, 5-Nov-2021.) |
⊢ (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸) | ||
Theorem | wlkvtxeledg 29656* | Each pair of adjacent vertices in a walk is a subset of an edge. (Contributed by AV, 28-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) | ||
Theorem | wlkvtxiedg 29657* | The vertices of a walk are connected by indexed edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘𝐹))∃𝑒 ∈ ran 𝐼{(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ 𝑒) | ||
Theorem | relwlk 29658 | The set (Walks‘𝐺) of all walks on 𝐺 is a set of pairs by our definition of a walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 19-Feb-2021.) |
⊢ Rel (Walks‘𝐺) | ||
Theorem | wlkvv 29659 | If there is at least one walk in the graph, all walks are in the universal class of ordered pairs. (Contributed by AV, 2-Jan-2021.) |
⊢ ((1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊) → 𝑊 ∈ (V × V)) | ||
Theorem | wlkop 29660 | A walk is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 1-Jan-2021.) |
⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | ||
Theorem | wlkcpr 29661 | A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊)) | ||
Theorem | wlk2f 29662* | If there is a walk 𝑊 there is a pair of functions representing this walk. (Contributed by Alexander van der Vekens, 22-Jul-2018.) |
⊢ (𝑊 ∈ (Walks‘𝐺) → ∃𝑓∃𝑝 𝑓(Walks‘𝐺)𝑝) | ||
Theorem | wlkcomp 29663* | A walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = (1st ‘𝑊) & ⊢ 𝑃 = (2nd ‘𝑊) ⇒ ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (Walks‘𝐺) ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))) | ||
Theorem | wlkcompim 29664* | Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = (1st ‘𝑊) & ⊢ 𝑃 = (2nd ‘𝑊) ⇒ ⊢ (𝑊 ∈ (Walks‘𝐺) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))) | ||
Theorem | wlkelwrd 29665 | The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = (1st ‘𝑊) & ⊢ 𝑃 = (2nd ‘𝑊) ⇒ ⊢ (𝑊 ∈ (Walks‘𝐺) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) | ||
Theorem | wlkeq 29666* | Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.) (Revised by AV, 14-Apr-2021.) |
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥)))) | ||
Theorem | edginwlk 29667 | The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 9-Dec-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((Fun 𝐼 ∧ 𝐹 ∈ Word dom 𝐼 ∧ 𝐾 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝐾)) ∈ 𝐸) | ||
Theorem | upgredginwlk 29668 | The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom 𝐼) → (𝐾 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝐾)) ∈ 𝐸)) | ||
Theorem | iedginwlk 29669 | The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 23-Apr-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((Fun 𝐼 ∧ 𝐹(Walks‘𝐺)𝑃 ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑋)) ∈ ran 𝐼) | ||
Theorem | wlkl1loop 29670 | A walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.) |
⊢ (((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺)) | ||
Theorem | wlk1walk 29671* | A walk is a 1-walk "on the edge level" according to Aksoy et al. (Contributed by AV, 30-Dec-2020.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (1..^(♯‘𝐹))1 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))))) | ||
Theorem | wlk1ewlk 29672 | A walk is an s-walk "on the edge level" (with s=1) according to Aksoy et al. (Contributed by AV, 5-Jan-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ (𝐺 EdgWalks 1)) | ||
Theorem | upgriswlk 29673* | Properties of a pair of functions to be a walk in a pseudograph. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 28-Oct-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | upgrwlkedg 29674* | The edges of a walk in a pseudograph join exactly the two corresponding adjacent vertices in the walk. (Contributed by AV, 27-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) | ||
Theorem | upgrwlkcompim 29675* | Implications for the properties of the components of a walk in a pseudograph. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 14-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = (1st ‘𝑊) & ⊢ 𝑃 = (2nd ‘𝑊) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) | ||
Theorem | wlkvtxedg 29676* | The vertices of a walk are connected by edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘𝐹))∃𝑒 ∈ 𝐸 {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ 𝑒) | ||
Theorem | upgrwlkvtxedg 29677* | The pairs of connected vertices of a walk are edges in a pseudograph. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸) | ||
Theorem | uspgr2wlkeq 29678* | Conditions for two walks within the same simple pseudograph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.) (Revised by AV, 14-Apr-2021.) |
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)))) | ||
Theorem | uspgr2wlkeq2 29679 | Conditions for two walks within the same simple pseudograph to be identical. It is sufficient that the vertices (in the same order) are identical. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.) |
⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘𝐵)) = 𝑁)) → ((2nd ‘𝐴) = (2nd ‘𝐵) → 𝐴 = 𝐵)) | ||
Theorem | uspgr2wlkeqi 29680 | Conditions for two walks within the same simple pseudograph to be identical. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 6-May-2021.) |
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)) → 𝐴 = 𝐵) | ||
Theorem | umgrwlknloop 29681* | In a multigraph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 3-Jan-2021.) |
⊢ ((𝐺 ∈ UMGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) | ||
Theorem | wlkResOLD 29682* | Obsolete version of opabresex2 7484 as of 13-Dec-2024. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 30-Dec-2020.) (Proof shortened by AV, 15-Jan-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑓(𝑊‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝) ⇒ ⊢ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑊‘𝐺)𝑝 ∧ 𝜑)} ∈ V | ||
Theorem | wlkv0 29683 | If there is a walk in the null graph (a class without vertices), it would be the pair consisting of empty sets. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.) |
⊢ (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (Walks‘𝐺)) → ((1st ‘𝑊) = ∅ ∧ (2nd ‘𝑊) = ∅)) | ||
Theorem | g0wlk0 29684 | There is no walk in a null graph (a class without vertices). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.) |
⊢ ((Vtx‘𝐺) = ∅ → (Walks‘𝐺) = ∅) | ||
Theorem | 0wlk0 29685 | There is no walk for the empty set, i.e. in a null graph. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.) |
⊢ (Walks‘∅) = ∅ | ||
Theorem | wlk0prc 29686 | There is no walk in a null graph (a class without vertices). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.) |
⊢ ((𝑆 ∉ V ∧ (Vtx‘𝑆) = (Vtx‘𝐺)) → (Walks‘𝐺) = ∅) | ||
Theorem | wlklenvclwlk 29687 | The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by JJ, 14-Jan-2024.) |
⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (〈𝐹, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (Walks‘𝐺) → (♯‘𝐹) = (♯‘𝑊))) | ||
Theorem | wlkson 29688* | The set of walks between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 30-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)}) | ||
Theorem | iswlkon 29689 | 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.) (Revised by AV, 31-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) | ||
Theorem | wlkonprop 29690 | Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 31-Dec-2020.) (Proof shortened by AV, 16-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) | ||
Theorem | wlkpvtx 29691 | A walk connects vertices. (Contributed by AV, 22-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑁 ∈ (0...(♯‘𝐹)) → (𝑃‘𝑁) ∈ 𝑉)) | ||
Theorem | wlkepvtx 29692 | The endpoints of a walk are vertices. (Contributed by AV, 31-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘(♯‘𝐹)) ∈ 𝑉)) | ||
Theorem | wlkoniswlk 29693 | A walk between two vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 2-Jan-2021.) |
⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | wlkonwlk 29694 | A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃) | ||
Theorem | wlkonwlk1l 29695 | A walk is a walk from its first vertex to its last vertex. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 22-Mar-2021.) |
⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) ⇒ ⊢ (𝜑 → 𝐹((𝑃‘0)(WalksOn‘𝐺)(lastS‘𝑃))𝑃) | ||
Theorem | wlksoneq1eq2 29696 | Two walks with identical sequences of vertices start and end at the same vertices. (Contributed by AV, 14-May-2021.) |
⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐻(𝐶(WalksOn‘𝐺)𝐷)𝑃) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | wlkonl1iedg 29697* | If there is a walk between two vertices 𝐴 and 𝐵 at least of length 1, then the start vertex 𝐴 is incident with an edge. (Contributed by AV, 4-Apr-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ (♯‘𝐹) ≠ 0) → ∃𝑒 ∈ ran 𝐼 𝐴 ∈ 𝑒) | ||
Theorem | wlkon2n0 29698 | The length of a walk between two different vertices is not 0 (i.e. is at least 1). (Contributed by AV, 3-Apr-2021.) |
⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐴 ≠ 𝐵) → (♯‘𝐹) ≠ 0) | ||
Theorem | 2wlklem 29699* | Lemma for theorems for walks of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
⊢ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) | ||
Theorem | upgr2wlk 29700 | Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 28-Oct-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |