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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}”⟩
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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 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