 Home Metamath Proof ExplorerTheorem List (p. 271 of 424) < 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: Metamath Proof Explorer (1-27759) Hilbert Space Explorer (27760-29284) Users' Mathboxes (29285-42322)

Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremupgr1trld 27001 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 27002 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 27003 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 27004 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 27005 A loop (which is an edge at index 𝐽) induces a cycle of length 1 in a hypergraph. (Contributed by AV, 2-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ⟨“𝐽”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩)

Theorem1pthon2v 27006* 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 27007* 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)

Theoremwlk2v2elem1 27008 Lemma 1 for wlk2v2e 27010: 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

Theoremwlk2v2elem2 27009* Lemma 2 for wlk2v2e 27010: 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))}

Theoremwlk2v2e 27010 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(Walks‘𝐺)𝑃

Theoremntrl2v2e 27011 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 wlk2v2e 27010, 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.) (Proof shortened by AV, 30-Oct-2021.)
𝐼 = ⟨“{𝑋, 𝑌}”⟩    &   𝐹 = ⟨“00”⟩    &   𝑋 ∈ V    &   𝑌 ∈ V    &   𝑃 = ⟨“𝑋𝑌𝑋”⟩    &   𝐺 = ⟨{𝑋, 𝑌}, 𝐼        ¬ 𝐹(Trails‘𝐺)𝑃

Theorem3wlkdlem1 27012 Lemma 1 for 3wlkd 27023. (Contributed by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩       (#‘𝑃) = ((#‘𝐹) + 1)

Theorem3wlkdlem2 27013 Lemma 2 for 3wlkd 27023. (Contributed by AV, 7-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩    &   𝐹 = ⟨“𝐽𝐾𝐿”⟩       (0..^(#‘𝐹)) = {0, 1, 2}

Theorem3wlkdlem3 27014 Lemma 3 for 3wlkd 27023. (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)))

Theorem3wlkdlem4 27015* Lemma 4 for 3wlkd 27023. (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)

Theorem3wlkdlem5 27016* Lemma 5 for 3wlkd 27023. (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 27017* Lemma 1 for 3pthd 27027. (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)))

Theorem3wlkdlem6 27018 Lemma 6 for 3wlkd 27023. (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)))

Theorem3wlkdlem7 27019 Lemma 7 for 3wlkd 27023. (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))

Theorem3wlkdlem8 27020 Lemma 8 for 3wlkd 27023. (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))

Theorem3wlkdlem9 27021 Lemma 9 for 3wlkd 27023. (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))))

Theorem3wlkdlem10 27022* Lemma 10 for 3wlkd 27023. (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)))

Theorem3wlkd 27023 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(Walks‘𝐺)𝑃)

Theorem3wlkond 27024 A 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 27025 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.) (Proof shortened by AV, 30-Oct-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 27026 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 27027 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 27028 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 27029 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.) (Proof shortened by AV, 30-Oct-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 27030 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 27031 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.) (Proof shortened by AV, 30-Oct-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 27032 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 27033* 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 27034 Lemma for uhgr3cyclex 27035. (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 27035* 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 27036* 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 27037* 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 27038* 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))))

16.3.12  Connected graphs

Syntaxcconngr 27039 Extend class notation with connected graphs.
class ConnGraph

Definitiondf-conngr 27040* 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 26983 and dfconngr1 27041. (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 27041* 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 27042* 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 27043* 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 27044 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 27045 A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
∅ ∈ ConnGraph

Theorem0vconngr 27046 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 27047 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 27048* A vertex in a connected graph with more than one vertex is incident with at least one edge. Formerly part of proof for vdgn0frgrv2 27152. (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 27049 A vertex in a connected multigraph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (((𝐺 ∈ ConnGraph ∧ 𝐺 ∈ UMGraph ) ∧ (𝑁𝑉 ∧ 1 < (#‘𝑉))) → ((VtxDeg‘𝐺)‘𝑁) ≠ 0)

16.4  Eulerian paths and the Konigsberg Bridge problem

16.4.1  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 27051 resp. iseupth 27054. (EulerPaths‘𝐺) is the set of all Eulerian paths in graph 𝐺, see eupths 27053. 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 27051, 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 27070). Vice versa, removing one edge from a graph with an Eulerian circuit results in a graph with an Eulerian path, see eucrct2eupth 27098.

An Eulerian path does not have to be a path in the meaning of definition df-pths 26606, 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 27094, it is shown that a pseudograph with an Eulerian path has either zero or two vertices of odd degree, and eulercrct 27095 shows that a pseudograph with an Eulerian circuit has only vertices of even degree.

Syntaxceupth 27050 Extend class notation with Eulerian paths.
class EulerPaths

Definitiondf-eupth 27051* 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 27052 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 27053* The Eulerian paths on the graph 𝐺. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.)
𝐼 = (iEdg‘𝐺)       (EulerPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝𝑓:(0..^(#‘𝑓))–onto→dom 𝐼)}

Theoremiseupth 27054 The property "𝐹, 𝑃 is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃𝐹:(0..^(#‘𝐹))–onto→dom 𝐼))

Theoremiseupthf1o 27055 The property "𝐹, 𝑃 is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼))

Theoremeupthi 27056 Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼))

Theoremeupthf1o 27057 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 27058 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 27059 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 27060* The property "𝐹, 𝑃 is an Eulerian path on the pseudograph 𝐺". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)
𝐼 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       (𝐺 ∈ UPGraph → (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremupgreupthi 27061* Properties of an Eulerian path in a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝐼 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))

Theoremupgreupthseg 27062 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 27063 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 27064 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)

Theoremeupthiswlk 27065 An Eulerian path is a walk. (Contributed by AV, 6-Apr-2021.)
(𝐹(EulerPaths‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)

Theoremeupthpf 27066 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 27067 There is an Eulerian path on an empty graph, i.e. a graph with at least one vertex, but without an edge. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 5-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐴𝑉𝐼 = ∅) → ∅(EulerPaths‘𝐺){⟨0, 𝐴⟩})

Theoremeupthres 27068 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.) (Proof shortened by AV, 30-Oct-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 27069 Append one path segment to an Eulerian path 𝐹, 𝑃 to become an Eulerian path 𝐻, 𝑄 of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 7-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (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 27070 Append one path segment to an Eulerian path 𝐹, 𝑃 which may not be an (Eulerian) circuit to become an Eulerian circuit 𝐻, 𝑄 of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. (Contributed by AV, 11-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (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 27071 Lemma for eupth2 27092. (Contributed by Mario Carneiro, 8-Apr-2015.)
(𝑈𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))

Theoremeupth2lem2 27072 Lemma for eupth2 27092. (Contributed by Mario Carneiro, 8-Apr-2015.)
𝐵 ∈ V       ((𝐵𝐶𝐵 = 𝑈) → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))

Theoremtrlsegvdeglem1 27073 Lemma for trlsegvdeg 27080. (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 27074 Lemma for trlsegvdeg 27080. (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 27075 Lemma for trlsegvdeg 27080. (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 27076 Lemma for trlsegvdeg 27080. (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 27077 Lemma for trlsegvdeg 27080. (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 27078 Lemma for trlsegvdeg 27080. (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 27079 Lemma for trlsegvdeg 27080. (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 27080 Formerly part of proof of eupth2lem3 27089: If a trail in a graph 𝐺 induces a subgraph 𝑍 with the vertices 𝑉 of 𝐺 and the edges being the edges of the walk, and a subgraph 𝑋 with the vertices 𝑉 of 𝐺 and the edges being the edges of the walk except the last one, and a subgraph 𝑌 with the vertices 𝑉 of 𝐺 and one edges being the last edge of the walk, then the vertex degree of any vertex 𝑈 of 𝐺 within 𝑍 is the sum of the vertex degree of 𝑈 within 𝑋 and the vertex degree of 𝑈 within 𝑌. Note that this theorem would not hold for arbitrary walks (if the last edge was identical with a previous edge, the degree of the vertices incident with this edge would not be increased because of this edge). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (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)))

Theoremeupth2lem3lem1 27081 Lemma for eupth2lem3 27089. (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 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ0)

Theoremeupth2lem3lem2 27082 Lemma for eupth2lem3 27089. (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 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℕ0)

Theoremeupth2lem3lem3 27083* Lemma for eupth2lem3 27089, formerly part of proof of eupth2lem3 27089: If a loop {(𝑃𝑁), (𝑃‘(𝑁 + 1))} is added to a trail, the degree of the vertices with odd degree remains odd (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 21-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (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 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))    &   (𝜑 → if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))))       ((𝜑 ∧ (𝑃𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theoremeupth2lem3lem4 27084* Lemma for eupth2lem3 27089, formerly part of proof of eupth2lem3 27089: If an edge (not a loop) is added to a trail, the degree of the end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (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 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))    &   (𝜑 → if-((𝑃𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹𝑁)) = {(𝑃𝑁)}, {(𝑃𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹𝑁))))    &   (𝜑 → (𝐼‘(𝐹𝑁)) ∈ 𝒫 𝑉)       ((𝜑 ∧ (𝑃𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 = (𝑃𝑁) ∨ 𝑈 = (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theoremeupth2lem3lem5 27085* Lemma for eupth2 27092. (Contributed by AV, 25-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 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))    &   (𝜑 → (𝐼‘(𝐹𝑁)) = {(𝑃𝑁), (𝑃‘(𝑁 + 1))})       (𝜑 → (𝐼‘(𝐹𝑁)) ∈ 𝒫 𝑉)

Theoremeupth2lem3lem6 27086* Formerly part of proof of eupth2lem3 27089: If an edge (not a loop) is added to a trail, the degree of vertices not being end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). Remark: This seems to be not valid for hyperedges joining more vertices than (𝑃‘0) and (𝑃𝑁): if there is a third vertex in the edge, and this vertex is already contained in the trail, then the degree of this vertex could be affected by this edge! (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (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 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))    &   (𝜑 → (𝐼‘(𝐹𝑁)) = {(𝑃𝑁), (𝑃‘(𝑁 + 1))})       ((𝜑 ∧ (𝑃𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theoremeupth2lem3lem7 27087* Lemma for eupth2lem3 27089: Combining trlsegvdeg 27080, eupth2lem3lem3 27083, eupth2lem3lem4 27084 and eupth2lem3lem6 27086. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 27-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 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))    &   (𝜑 → (𝐼‘(𝐹𝑁)) = {(𝑃𝑁), (𝑃‘(𝑁 + 1))})       (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑍)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theoremeupthvdres 27088 Formerly part of proof of eupth2 27092: The vertex degree remains the same for all vertices if the edges are restricted to the edges of an Eulerian path. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐺𝑊)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)    &   𝐻 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(#‘𝐹))))⟩       (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))

Theoremeupth2lem3 27089* Lemma for eupth2 27092. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐺 ∈ UPGraph )    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)    &   𝐻 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩    &   𝑋 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → (𝑁 + 1) ≤ (#‘𝐹))    &   (𝜑𝑈𝑉)    &   (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐻)‘𝑥)} = if((𝑃‘0) = (𝑃𝑁), ∅, {(𝑃‘0), (𝑃𝑁)}))       (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theoremeupth2lemb 27090* Lemma for eupth2 27092 (induction basis): There are no vertices of odd degree in an Eulerian path of length 0, having no edge and identical endpoints (the single vertex of the Eulerian path). Formerly part of proof for eupth2 27092. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐺 ∈ UPGraph )    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)       (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = ∅)

Theoremeupth2lems 27091* Lemma for eupth2 27092 (induction step): The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct, if the Eulerian path shortened by one edge has this property. Formerly part of proof for eupth2 27092. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐺 ∈ UPGraph )    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)       ((𝜑𝑛 ∈ ℕ0) → ((𝑛 ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃𝑛), ∅, {(𝑃‘0), (𝑃𝑛)})) → ((𝑛 + 1) ≤ (#‘𝐹) → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))

Theoremeupth2 27092* The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐺 ∈ UPGraph )    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)       (𝜑 → {𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))

Theoremeulerpathpr 27093* A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})

Theoremeulerpath 27094* A pseudograph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UPGraph ∧ (EulerPaths‘𝐺) ≠ ∅) → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})

Theoremeulercrct 27095* A pseudograph with an Eulerian circuit 𝐹, 𝑃 (an "Eulerian pseudograph") has only vertices of even degree. (Contributed by AV, 12-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃𝐹(Circuits‘𝐺)𝑃) → ∀𝑥𝑉 2 ∥ ((VtxDeg‘𝐺)‘𝑥))

Theoremeucrctshift 27096* Cyclically shifting the indices of an Eulerian circuit 𝐹, 𝑃 results in an Eulerian circuit 𝐻, 𝑄. (Contributed by AV, 15-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)       (𝜑 → (𝐻(EulerPaths‘𝐺)𝑄𝐻(Circuits‘𝐺)𝑄))

Theoremeucrct2eupth1 27097 Removing one edge (𝐼‘(𝐹𝑁)) from a nonempty graph 𝐺 with an Eulerian circuit 𝐹, 𝑃 results in a graph 𝑆 with an Eulerian path 𝐻, 𝑄. This is the special case of eucrct2eupth 27098 (with 𝐽 = (𝑁 − 1)) where the last segment/edge of the circuit is removed. (Contributed by AV, 11-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   (Vtx‘𝑆) = 𝑉    &   (𝜑 → 0 < (#‘𝐹))    &   (𝜑𝑁 = ((#‘𝐹) − 1))    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   𝐻 = (𝐹 ↾ (0..^𝑁))    &   𝑄 = (𝑃 ↾ (0...𝑁))       (𝜑𝐻(EulerPaths‘𝑆)𝑄)

Theoremeucrct2eupth 27098* Removing one edge (𝐼‘(𝐹𝐽)) from a graph 𝐺 with an Eulerian circuit 𝐹, 𝑃 results in a graph 𝑆 with an Eulerian path 𝐻, 𝑄. (Contributed by AV, 17-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(EulerPaths‘𝐺)𝑃)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   (Vtx‘𝑆) = 𝑉    &   (𝜑𝑁 = (#‘𝐹))    &   (𝜑𝐽 ∈ (0..^𝑁))    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ ((0..^𝑁) ∖ {𝐽}))))    &   𝐾 = (𝐽 + 1)    &   𝐻 = ((𝐹 cyclShift 𝐾) ↾ (0..^(𝑁 − 1)))    &   𝑄 = (𝑥 ∈ (0..^𝑁) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁))))       (𝜑𝐻(EulerPaths‘𝑆)𝑄)

16.4.2  The Königsberg Bridge problem

According to Wikipedia ("Seven Bridges of Königsberg", 9-Mar-2021, https://en.wikipedia.org/wiki/Seven_Bridges_of_Koenigsberg): "The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology. The city of Königsberg in [East] Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands - Kneiphof and Lomse - which were connected to each other, or to the two mainland portions of the city, by seven bridges. The problem was to devise a walk through the city that would cross each of those bridges once and only once.". Euler proved that the problem has no solution by applying Euler's theorem to the Königsberg graph, which is obtained by replacing each land mass with an abstract "vertex" or node, and each bridge with an abstract connection, an "edge", which connects two land masses/vertices. The Königsberg graph 𝐺 is a multigraph consisting of 4 vertices and 7 edges, represented by the following ordered pair: 𝐺 = ⟨(0...3), ⟨“{0, 1}{0, 2} {0, 3}{1, 2}{1, 2}{2, 3}{2, 3}”⟩⟩, see konigsbergumgr 27105. konigsberg 27112 shows that the Königsberg graph has no Eulerian path, thus the Königsberg Bridge problem has no solution.

Theoremkonigsbergvtx 27099 The set of vertices of the Königsberg graph 𝐺. (Contributed by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       (Vtx‘𝐺) = (0...3)

Theoremkonigsbergiedg 27100 The indexed edges of the Königsberg graph 𝐺. (Contributed by AV, 28-Feb-2021.)
𝑉 = (0...3)    &   𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸       (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42322
 Copyright terms: Public domain < Previous  Next >