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Theorem List for Metamath Proof Explorer - 41401-41500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0pthon1-av 41401 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉 → ∅(𝑁(PathsOn‘𝐺)𝑁){⟨0, 𝑁⟩})
 
Theorem0pthonv-av 41402* For each vertex there is a path of length 0 from the vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 21-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉 → ∃𝑓𝑝 𝑓(𝑁(PathsOn‘𝐺)𝑁)𝑝)
 
Theorem0clWlk 41403 A pair of an empty set (of edges) and a second set (of vertices) is a closed walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 17-Feb-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺𝑋𝑃𝑍) → (∅(ClWalkS‘𝐺)𝑃𝑃:(0...0)⟶𝑉))
 
Theorem0clwlk0 41404 There is no closed walk in the empty set (i.e. the null graph). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
(ClWalkS‘∅) = ∅
 
Theorem0Crct 41405 A pair of an empty set (of edges) and a second set (of vertices) is a circuit if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
((𝐺𝑊𝑃𝑍) → (∅(CircuitS‘𝐺)𝑃𝑃:(0...0)⟶(Vtx‘𝐺)))
 
Theorem0Cycl 41406 A pair of an empty set (of edges) and a second set (of vertices) is a cycle if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
((𝐺𝑊𝑃𝑍) → (∅(CycleS‘𝐺)𝑃𝑃:(0...0)⟶(Vtx‘𝐺)))
 
Theorem1pthdlem1 41407 Lemma 1 for 1pthd 41415. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩       Fun (𝑃 ↾ (1..^(#‘𝐹)))
 
Theorem1pthdlem2 41408 Lemma 2 for 1pthd 41415. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩       ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅
 
Theorem11wlkdlem1 41409 Lemma 1 for 11wlkd 41413. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑𝑃:(0...(#‘𝐹))⟶𝑉)
 
Theorem11wlkdlem2 41410 Lemma 2 for 11wlkd 41413. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))       (𝜑𝑋 ∈ (𝐼𝐽))
 
Theorem11wlkdlem3 41411 Lemma 3 for 11wlkd 41413. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))       (𝜑𝐹 ∈ Word dom 𝐼)
 
Theorem11wlkdlem4 41412* Lemma 4 for 11wlkd 41413. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))       (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
 
Theorem11wlkd 41413 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a walk. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(1Walks‘𝐺)𝑃)
 
Theorem1trld 41414 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(TrailS‘𝐺)𝑃)
 
Theorem1pthd 41415 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(PathS‘𝐺)𝑃)
 
Theorem1pthond 41416 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑋 = 𝑌) → (𝐼𝐽) = {𝑋})    &   ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ (𝐼𝐽))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃)
 
Theoremupgr11wlkdlem1 41417 Lemma 1 for upgr11wlkd 41419. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋 ∈ (Vtx‘𝐺))    &   (𝜑𝑌 ∈ (Vtx‘𝐺))    &   (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})       ((𝜑𝑋 = 𝑌) → ((iEdg‘𝐺)‘𝐽) = {𝑋})
 
Theoremupgr11wlkdlem2 41418 Lemma 2 for upgr11wlkd 41419. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋 ∈ (Vtx‘𝐺))    &   (𝜑𝑌 ∈ (Vtx‘𝐺))    &   (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})       ((𝜑𝑋𝑌) → {𝑋, 𝑌} ⊆ ((iEdg‘𝐺)‘𝐽))
 
Theoremupgr11wlkd 41419 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a walk. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋 ∈ (Vtx‘𝐺))    &   (𝜑𝑌 ∈ (Vtx‘𝐺))    &   (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})    &   (𝜑𝐺 ∈ UPGraph )       (𝜑𝐹(1Walks‘𝐺)𝑃)
 
Theoremupgr1trld 41420 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋 ∈ (Vtx‘𝐺))    &   (𝜑𝑌 ∈ (Vtx‘𝐺))    &   (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})    &   (𝜑𝐺 ∈ UPGraph )       (𝜑𝐹(TrailS‘𝐺)𝑃)
 
Theoremupgr1pthd 41421 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋 ∈ (Vtx‘𝐺))    &   (𝜑𝑌 ∈ (Vtx‘𝐺))    &   (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})    &   (𝜑𝐺 ∈ UPGraph )       (𝜑𝐹(PathS‘𝐺)𝑃)
 
Theoremupgr1pthond 41422 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by AV, 22-Jan-2021.)
𝑃 = ⟨“𝑋𝑌”⟩    &   𝐹 = ⟨“𝐽”⟩    &   (𝜑𝑋 ∈ (Vtx‘𝐺))    &   (𝜑𝑌 ∈ (Vtx‘𝐺))    &   (𝜑 → ((iEdg‘𝐺)‘𝐽) = {𝑋, 𝑌})    &   (𝜑𝐺 ∈ UPGraph )       (𝜑𝐹(𝑋(PathsOn‘𝐺)𝑌)𝑃)
 
Theoremlppthon 41423 A loop (which is an edge at index 𝐽) induces a path of length 1 from a vertex to itself in a hypergraph. (Contributed by AV, 1-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ⟨“𝐽”⟩(𝐴(PathsOn‘𝐺)𝐴)⟨“𝐴𝐴”⟩)
 
Theoremlp1cycl 41424 A loop (which is an edge at index 𝐽) induces a cycle of length 1 in a hypergraph. (Contributed by AV, 2-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ⟨“𝐽”⟩(CycleS‘𝐺)⟨“𝐴𝐴”⟩)
 
Theorem1pthon2v-av 41425* For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)
 
Theorem1pthon2ve 41426* For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Proof shortened by AV, 15-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ (𝐴𝑉𝐵𝑉) ∧ {𝐴, 𝐵} ∈ 𝐸) → ∃𝑓𝑝 𝑓(𝐴(PathsOn‘𝐺)𝐵)𝑝)
 
Theorem1wlk2v2elem1 41427 Lemma 1 for 1wlk2v2e 41429: 𝐹 is a length 2 word of over {0}, the domain of the singleton word 𝐼. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.)
𝐼 = ⟨“{𝑋, 𝑌}”⟩    &   𝐹 = ⟨“00”⟩       𝐹 ∈ Word dom 𝐼
 
Theorem1wlk2v2elem2 41428* Lemma 2 for 1wlk2v2e 41429: The values of 𝐼 after 𝐹 are edges between two vertices enumerated by 𝑃. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.)
𝐼 = ⟨“{𝑋, 𝑌}”⟩    &   𝐹 = ⟨“00”⟩    &   𝑋 ∈ V    &   𝑌 ∈ V    &   𝑃 = ⟨“𝑋𝑌𝑋”⟩       𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}
 
Theorem1wlk2v2e 41429 In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk. Notice that 𝐺 is a simple graph (without loops) only if 𝑋𝑌. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
𝐼 = ⟨“{𝑋, 𝑌}”⟩    &   𝐹 = ⟨“00”⟩    &   𝑋 ∈ V    &   𝑌 ∈ V    &   𝑃 = ⟨“𝑋𝑌𝑋”⟩    &   𝐺 = ⟨{𝑋, 𝑌}, 𝐼       𝐹(1Walks‘𝐺)𝑃
 
Theoremntrl2v2e 41430 A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk, see 1wlk2v2e 41429, but not a trail. Notice that 𝐺 is a simple graph (without loops) only if 𝑋𝑌. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
𝐼 = ⟨“{𝑋, 𝑌}”⟩    &   𝐹 = ⟨“00”⟩    &   𝑋 ∈ V    &   𝑌 ∈ V    &   𝑃 = ⟨“𝑋𝑌𝑋”⟩    &   𝐺 = ⟨{𝑋, 𝑌}, 𝐼        ¬ 𝐹(TrailS‘𝐺)𝑃
 
Theorem31wlkdlem1 41431 Lemma 1 for 31wlkd 41442. (Contributed by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩       (#‘𝑃) = ((#‘𝐹) + 1)
 
Theorem31wlkdlem2 41432 Lemma 2 for 31wlkd 41442. (Contributed by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩       (0..^(#‘𝐹)) = {0, 1, 2}
 
Theorem31wlkdlem3 41433 Lemma 3 for 31wlkd 41442. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))       (𝜑 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐷)))
 
Theorem31wlkdlem4 41434* Lemma 4 for 31wlkd 41442. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))       (𝜑 → ∀𝑘 ∈ (0...(#‘𝐹))(𝑃𝑘) ∈ 𝑉)
 
Theorem31wlkdlem5 41435* Lemma 5 for 31wlkd 41442. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))       (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
 
Theorem3pthdlem1 41436* Lemma 1 for 3pthd 41446. (Contributed by AV, 9-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))       (𝜑 → ∀𝑘 ∈ (0..^(#‘𝑃))∀𝑗 ∈ (1..^(#‘𝐹))(𝑘𝑗 → (𝑃𝑘) ≠ (𝑃𝑗)))
 
Theorem31wlkdlem6 41437 Lemma 6 for 31wlkd 41442. (Contributed by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))       (𝜑 → (𝐴 ∈ (𝐼𝐽) ∧ 𝐵 ∈ (𝐼𝐾) ∧ 𝐶 ∈ (𝐼𝐿)))
 
Theorem31wlkdlem7 41438 Lemma 7 for 31wlkd 41442. (Contributed by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))       (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V))
 
Theorem31wlkdlem8 41439 Lemma 8 for 31wlkd 41442. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))       (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾 ∧ (𝐹‘2) = 𝐿))
 
Theorem31wlkdlem9 41440 Lemma 9 for 31wlkd 41442. (Contributed by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))       (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ∧ {𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1)) ∧ {𝐶, 𝐷} ⊆ (𝐼‘(𝐹‘2))))
 
Theorem31wlkdlem10 41441* Lemma 10 for 31wlkd 41442. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))       (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))
 
Theorem31wlkd 41442 Construction of a walk from two given edges in a graph. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(1Walks‘𝐺)𝑃)
 
Theorem31wlkond 41443 A 1-walk of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(𝐴(WalksOn‘𝐺)𝐷)𝑃)
 
Theorem3trld 41444 Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))       (𝜑𝐹(TrailS‘𝐺)𝑃)
 
Theorem3trlond 41445 A trail of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))       (𝜑𝐹(𝐴(TrailsOn‘𝐺)𝐷)𝑃)
 
Theorem3pthd 41446 A path of length 3 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))       (𝜑𝐹(PathS‘𝐺)𝑃)
 
Theorem3pthond 41447 A path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))       (𝜑𝐹(𝐴(PathsOn‘𝐺)𝐷)𝑃)
 
Theorem3spthd 41448 A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))    &   (𝜑𝐴𝐷)       (𝜑𝐹(SPathS‘𝐺)𝑃)
 
Theorem3spthond 41449 A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))    &   (𝜑𝐴𝐷)       (𝜑𝐹(𝐴(SPathsOn‘𝐺)𝐷)𝑃)
 
Theorem3cycld 41450 Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))    &   (𝜑𝐴 = 𝐷)       (𝜑𝐹(CycleS‘𝐺)𝑃)
 
Theorem3cyclpd 41451 Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩    &   (𝜑 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)))    &   (𝜑 → ((𝐴𝐵𝐴𝐶) ∧ (𝐵𝐶𝐵𝐷) ∧ 𝐶𝐷))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼𝐿)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → (𝐽𝐾𝐽𝐿𝐾𝐿))    &   (𝜑𝐴 = 𝐷)       (𝜑 → (𝐹(CycleS‘𝐺)𝑃 ∧ (#‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴))
 
Theoremupgr3v3e3cycl 41452* If there is a cycle of length 3 in a pseudograph, there are three distinct vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(CycleS‘𝐺)𝑃 ∧ (#‘𝐹) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐𝑐𝑎)))
 
Theoremuhgr3cyclexlem 41453 Lemma for uhgr3cyclex 41454. (Contributed by AV, 12-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((((𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ ((𝐽 ∈ dom 𝐼 ∧ {𝐵, 𝐶} = (𝐼𝐽)) ∧ (𝐾 ∈ dom 𝐼 ∧ {𝐶, 𝐴} = (𝐼𝐾)))) → 𝐽𝐾)
 
Theoremuhgr3cyclex 41454* If there are three different vertices in a hypergraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ ((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃𝑓𝑝(𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))
 
Theoremumgr3cyclex 41455* If there are three (different) vertices in a multigraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)) → ∃𝑓𝑝(𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))
 
Theoremumgr3v3e3cycl 41456* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ UMGraph → (∃𝑓𝑝(𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))
 
Theoremupgr4cycl4dv4e 41457* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(CycleS‘𝐺)𝑃 ∧ (#‘𝐹) = 4) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 ((({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸) ∧ ({𝑐, 𝑑} ∈ 𝐸 ∧ {𝑑, 𝑎} ∈ 𝐸)) ∧ ((𝑎𝑏𝑎𝑐𝑎𝑑) ∧ (𝑏𝑐𝑏𝑑𝑐𝑑))))
 
20.34.8.25  Connected graphs
 
Syntaxcconngr 41458 Extend class notation with connected graphs.
class ConnGraph
 
Definitiondf-conngr 41459* 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-av 41402 and dfconngr1 41460. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
ConnGraph = {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
 
Theoremdfconngr1 41460* 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.)
ConnGraph = {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛 ∈ (𝑣 ∖ {𝑘})∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
 
Theoremisconngr 41461* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘𝑉𝑛𝑉𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
 
Theoremisconngr1 41462* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ConnGraph ↔ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓𝑝 𝑓(𝑘(PathsOn‘𝐺)𝑛)𝑝))
 
Theoremcusconngr 41463 A complete hypergraph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 15-Feb-2021.)
((𝐺 ∈ UHGraph ∧ 𝐺 ∈ ComplGraph) → 𝐺 ∈ ConnGraph)
 
Theorem0conngr 41464 A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
∅ ∈ ConnGraph
 
Theorem0vconngr 41465 A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ ConnGraph)
 
Theorem1conngr 41466 A graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = {𝑁}) → 𝐺 ∈ ConnGraph)
 
Theoremconngrv2edg 41467* A vertex in a connected graph with more than one vertex is incident with at least one edge. Formerly part of proof for vdgn0frgrav2 26289. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ ConnGraph ∧ 𝑁𝑉 ∧ 1 < (#‘𝑉)) → ∃𝑒 ∈ ran 𝐼 𝑁𝑒)
 
Theoremvdn0conngrumgrv2 41468 A vertex in a connected multigraph with more than one vertex cannot have degree 0. Formerly vdgn0frgrav2 26289. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) ≠ 0)
 
20.34.8.26  Eulerian paths

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 41470 resp. iseupth 41473. (EulerPaths‘𝐺) is the set of all Eulerian paths in graph 𝐺, see eupths 41472. 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 41470, whereas there is no explicit definition for Eulerian circuits (yet): The statement "𝐹, 𝑃 is an Eulerian circuit" is formally expressed by (𝐹(EulerPaths‘𝐺)𝑃𝐹(CircuitS‘𝐺)𝑃).

Each Eulerian path can be made an Eulerian circuit by adding an edge which connects the endpoints of the Eulerian path (see eupth2eucrct 41490). Vice versa, removing one edge from a graph with an Eulerian circuit results in a graph with an Eulerian path, see eucrct2eupth 41518.

An Eulerian path does not have to be a path in the meaning of definition df-pths 41028, 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 41514, it is shown that a pseudograph with an Eulerian path has either zero or two vertices of odd degree, and eulercrct 41515 shows that a pseudograph with an Eulerian circuit has only vertices of even degree.

 
Syntaxceupth 41469 Extend class notation with Eulerian paths.
class EulerPaths
 
Definitiondf-eupth 41470* Define the set of all Eulerian paths on an arbitrary graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝑔)𝑝𝑓:(0..^(#‘𝑓))–onto→dom (iEdg‘𝑔))})
 
Theoremreleupth 41471 The set (EulerPaths‘𝐺) of all Eulerian paths on 𝐺 is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
Rel (EulerPaths‘𝐺)
 
Theoremeupths 41472* The Eulerian paths on the graph 𝐺. (Contributed by AV, 18-Feb-2021.)
𝐼 = (iEdg‘𝐺)       (𝐺𝑋 → (EulerPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝐺)𝑝𝑓:(0..^(#‘𝑓))–onto→dom 𝐼)})
 
Theoremiseupth 41473 The property "𝐹, 𝑃 is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺𝑋𝐹𝑌𝑃𝑍) → (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(TrailS‘𝐺)𝑃𝐹:(0..^(#‘𝐹))–onto→dom 𝐼)))
 
Theoremiseupthf1o 41474 The property "𝐹, 𝑃 is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺𝑋𝐹𝑌𝑃𝑍) → (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼)))
 
Theoremeupthtrli 41475 Properties of an Eulerian path as a trail. (Contributed by AV, 18-Feb-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(TrailS‘𝐺)𝑃𝐹:(0..^(#‘𝐹))–onto→dom 𝐼))
 
Theoremeupthi 41476 Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(1Walks‘𝐺)𝑃𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼))
 
Theoremeupthf1o 41477 The 𝐹 function in an Eulerian path is a bijection from a half-open range of nonnegative integers to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(EulerPaths‘𝐺)𝑃𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼)
 
Theoremeupthfi 41478 Any graph with an Eulerian path is of finite size, i.e. with a finite number of edges. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 18-Feb-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(EulerPaths‘𝐺)𝑃 → dom 𝐼 ∈ Fin)
 
Theoremeupthseg 41479 The 𝑁-th edge in an eulerian path is the edge having 𝑃(𝑁) and 𝑃(𝑁 + 1) as endpoints . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐹(EulerPaths‘𝐺)𝑃𝑁 ∈ (0..^(#‘𝐹))) → {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁)))
 
Theoremupgriseupth 41480* The property "𝐹, 𝑃 is an Eulerian path on the pseudograph 𝐺". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹𝑈𝑃𝑍) → (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
 
Theoremupgreupthi 41481* Properties of an Eulerian path in a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
 
Theoremupgreupthseg 41482 The 𝑁-th edge in an eulerian path is the edge from 𝑃(𝑁) to 𝑃(𝑁 + 1). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃𝑁 ∈ (0..^(#‘𝐹))) → (𝐼‘(𝐹𝑁)) = {(𝑃𝑁), (𝑃‘(𝑁 + 1))})
 
Theoremeupthcl 41483 An Eulerian path has length #(𝐹), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
(𝐹(EulerPaths‘𝐺)𝑃 → (#‘𝐹) ∈ ℕ0)
 
Theoremeupthistrl 41484 An Eulerian path is a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.) (Revised by AV, 18-Feb-2021.)
(𝐹(EulerPaths‘𝐺)𝑃𝐹(TrailS‘𝐺)𝑃)
 
Theoremeupthis1wlk 41485 An Eulerian path is a walk. (Contributed by AV, 6-Apr-2021.)
(𝐹(EulerPaths‘𝐺)𝑃𝐹(1Walks‘𝐺)𝑃)
 
Theoremeupthpf 41486 The 𝑃 function in an Eulerian path is a function from a finite sequence of nonnegative integers to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
(𝐹(EulerPaths‘𝐺)𝑃𝑃:(0...(#‘𝐹))⟶(Vtx‘𝐺))
 
Theoremeupth0 41487 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.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐴𝑉𝐼 = ∅) → ∅(EulerPaths‘𝐺){⟨0, 𝐴⟩})
 
Theoremeupthres 41488 The restriction 𝐻, 𝑄 of an Eulerian path 𝐹, 𝑃 to an initial segment of the path (of length 𝑁) forms an Eulerian path on the subgraph 𝑆 consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   𝐻 = (𝐹 ↾ (0..^𝑁))    &   𝑄 = (𝑃 ↾ (0...𝑁))    &   (Vtx‘𝑆) = 𝑉       (𝜑𝐻(EulerPaths‘𝑆)𝑄)
 
Theoremeupthp1 41489 Append one path segment to an Eulerian path 𝐹, 𝑃 to become an Eulerian path 𝐻, 𝑄 of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 7-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶𝑉)    &   (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐸 ∈ (Edg‘𝐺))    &   (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)    &   (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩})    &   𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   (Vtx‘𝑆) = 𝑉    &   ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {𝐶})       (𝜑𝐻(EulerPaths‘𝑆)𝑄)
 
Theoremeupth2eucrct 41490 Append one path segment to an Eulerian path 𝐹, 𝑃 which may not be an (Eulerian) circuit to become an Eulerian circuit 𝐻, 𝑄 of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. (Contributed by AV, 11-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶𝑉)    &   (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐸 ∈ (Edg‘𝐺))    &   (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)    &   (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩})    &   𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   (Vtx‘𝑆) = 𝑉    &   ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {𝐶})    &   (𝜑𝐶 = (𝑃‘0))       (𝜑 → (𝐻(EulerPaths‘𝑆)𝑄𝐻(CircuitS‘𝑆)𝑄))
 
Theoremeupth2lem1 41491 TODO-AV: Duplicate of eupath2lem1 26242! Lemma for eupath2 26245. (Contributed by Mario Carneiro, 8-Apr-2015.)
(𝑈𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
 
Theoremeupth2lem2 41492 TODO-AV: Duplicate of eupath2lem2 26243! Lemma for eupath2 26245. (Contributed by Mario Carneiro, 8-Apr-2015.)
𝐵 ∈ V       ((𝐵𝐶𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))
 
Theoremtrlsegvdeglem1 41493 Lemma for trlsegvdeg 41500. (Contributed by AV, 20-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(TrailS‘𝐺)𝑃)       (𝜑 → ((𝑃𝑁) ∈ 𝑉 ∧ (𝑃‘(𝑁 + 1)) ∈ 𝑉))
 
Theoremtrlsegvdeglem2 41494 Lemma for trlsegvdeg 41500. (Contributed by AV, 20-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(TrailS‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → Fun (iEdg‘𝑋))
 
Theoremtrlsegvdeglem3 41495 Lemma for trlsegvdeg 41500. (Contributed by AV, 20-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(TrailS‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → Fun (iEdg‘𝑌))
 
Theoremtrlsegvdeglem4 41496 Lemma for trlsegvdeg 41500. (Contributed by AV, 21-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(TrailS‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
 
Theoremtrlsegvdeglem5 41497 Lemma for trlsegvdeg 41500. (Contributed by AV, 21-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(TrailS‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → dom (iEdg‘𝑌) = {(𝐹𝑁)})
 
Theoremtrlsegvdeglem6 41498 Lemma for trlsegvdeg 41500. (Contributed by AV, 21-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(TrailS‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → dom (iEdg‘𝑋) ∈ Fin)
 
Theoremtrlsegvdeglem7 41499 Lemma for trlsegvdeg 41500. (Contributed by AV, 21-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(TrailS‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → dom (iEdg‘𝑌) ∈ Fin)
 
Theoremtrlsegvdeg 41500 Formerly part of proof of eupath2lem3 26244: If a trail in a graph 𝐺 induces a subgraph 𝑍 with the vertices 𝑉 of 𝐺 and the edges being the edges of the 1-walk, and a subgraph 𝑋 with the vertices 𝑉 of 𝐺 and the edges being the edges of the 1-walk except the last one, and a subgraph 𝑌 with the vertices 𝑉 of 𝐺 and one edges being the last edge of the 1-walk, then the vertex degree of any vertex 𝑈 of 𝐺 within 𝑍 is the sum of the vertex degree of 𝑈 within 𝑋 and the vertex degree of 𝑈 within 𝑌. Note that this theorem would not hold for arbitrary 1-walks (if the last edge was identical with a previous edge, the degree of the vertices incident with this edge would not be increased because of this edge). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑𝑈𝑉)    &   (𝜑𝐹(TrailS‘𝐺)𝑃)    &   (𝜑 → (Vtx‘𝑋) = 𝑉)    &   (𝜑 → (Vtx‘𝑌) = 𝑉)    &   (𝜑 → (Vtx‘𝑍) = 𝑉)    &   (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   (𝜑 → (iEdg‘𝑌) = {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩})    &   (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))       (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))
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