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Type | Label | Description |
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Statement | ||
Theorem | upgr1trld 29401 | 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.) |
β’ π = β¨βππββ© & β’ πΉ = β¨βπ½ββ© & β’ (π β π β (VtxβπΊ)) & β’ (π β π β (VtxβπΊ)) & β’ (π β ((iEdgβπΊ)βπ½) = {π, π}) & β’ (π β πΊ β UPGraph) β β’ (π β πΉ(TrailsβπΊ)π) | ||
Theorem | upgr1pthd 29402 | 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.) |
β’ π = β¨βππββ© & β’ πΉ = β¨βπ½ββ© & β’ (π β π β (VtxβπΊ)) & β’ (π β π β (VtxβπΊ)) & β’ (π β ((iEdgβπΊ)βπ½) = {π, π}) & β’ (π β πΊ β UPGraph) β β’ (π β πΉ(PathsβπΊ)π) | ||
Theorem | upgr1pthond 29403 | 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.) |
β’ π = β¨βππββ© & β’ πΉ = β¨βπ½ββ© & β’ (π β π β (VtxβπΊ)) & β’ (π β π β (VtxβπΊ)) & β’ (π β ((iEdgβπΊ)βπ½) = {π, π}) & β’ (π β πΊ β UPGraph) β β’ (π β πΉ(π(PathsOnβπΊ)π)π) | ||
Theorem | lppthon 29404 | A loop (which is an edge at index π½) induces a path of length 1 from a vertex to itself in a hypergraph. (Contributed by AV, 1-Feb-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β UHGraph β§ π½ β dom πΌ β§ (πΌβπ½) = {π΄}) β β¨βπ½ββ©(π΄(PathsOnβπΊ)π΄)β¨βπ΄π΄ββ©) | ||
Theorem | lp1cycl 29405 | A loop (which is an edge at index π½) induces a cycle of length 1 in a hypergraph. (Contributed by AV, 2-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β UHGraph β§ π½ β dom πΌ β§ (πΌβπ½) = {π΄}) β β¨βπ½ββ©(CyclesβπΊ)β¨βπ΄π΄ββ©) | ||
Theorem | 1pthon2v 29406* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ ((πΊ β UHGraph β§ (π΄ β π β§ π΅ β π) β§ βπ β πΈ {π΄, π΅} β π) β βπβπ π(π΄(PathsOnβπΊ)π΅)π) | ||
Theorem | 1pthon2ve 29407* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ ((πΊ β UHGraph β§ (π΄ β π β§ π΅ β π) β§ {π΄, π΅} β πΈ) β βπβπ π(π΄(PathsOnβπΊ)π΅)π) | ||
Theorem | wlk2v2elem1 29408 | Lemma 1 for wlk2v2e 29410: πΉ is a length 2 word of over {0}, the domain of the singleton word πΌ. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.) |
β’ πΌ = β¨β{π, π}ββ© & β’ πΉ = β¨β00ββ© β β’ πΉ β Word dom πΌ | ||
Theorem | wlk2v2elem2 29409* | Lemma 2 for wlk2v2e 29410: The values of πΌ after πΉ are edges between two vertices enumerated by π. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.) |
β’ πΌ = β¨β{π, π}ββ© & β’ πΉ = β¨β00ββ© & β’ π β V & β’ π β V & β’ π = β¨βπππββ© β β’ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))} | ||
Theorem | wlk2v2e 29410 | 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 πΊ is a simple graph (without loops) only if π β π. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.) |
β’ πΌ = β¨β{π, π}ββ© & β’ πΉ = β¨β00ββ© & β’ π β V & β’ π β V & β’ π = β¨βπππββ© & β’ πΊ = β¨{π, π}, πΌβ© β β’ πΉ(WalksβπΊ)π | ||
Theorem | ntrl2v2e 29411 | 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 29410, but not a trail. Notice that πΊ is a simple graph (without loops) only if π β π. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
β’ πΌ = β¨β{π, π}ββ© & β’ πΉ = β¨β00ββ© & β’ π β V & β’ π β V & β’ π = β¨βπππββ© & β’ πΊ = β¨{π, π}, πΌβ© β β’ Β¬ πΉ(TrailsβπΊ)π | ||
Theorem | 3wlkdlem1 29412 | Lemma 1 for 3wlkd 29423. (Contributed by AV, 7-Feb-2021.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© β β’ (β―βπ) = ((β―βπΉ) + 1) | ||
Theorem | 3wlkdlem2 29413 | Lemma 2 for 3wlkd 29423. (Contributed by AV, 7-Feb-2021.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© β β’ (0..^(β―βπΉ)) = {0, 1, 2} | ||
Theorem | 3wlkdlem3 29414 | Lemma 3 for 3wlkd 29423. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) β β’ (π β (((πβ0) = π΄ β§ (πβ1) = π΅) β§ ((πβ2) = πΆ β§ (πβ3) = π·))) | ||
Theorem | 3wlkdlem4 29415* | Lemma 4 for 3wlkd 29423. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) β β’ (π β βπ β (0...(β―βπΉ))(πβπ) β π) | ||
Theorem | 3wlkdlem5 29416* | Lemma 5 for 3wlkd 29423. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) β β’ (π β βπ β (0..^(β―βπΉ))(πβπ) β (πβ(π + 1))) | ||
Theorem | 3pthdlem1 29417* | Lemma 1 for 3pthd 29427. (Contributed by AV, 9-Feb-2021.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) β β’ (π β βπ β (0..^(β―βπ))βπ β (1..^(β―βπΉ))(π β π β (πβπ) β (πβπ))) | ||
Theorem | 3wlkdlem6 29418 | Lemma 6 for 3wlkd 29423. (Contributed by AV, 7-Feb-2021.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) β β’ (π β (π΄ β (πΌβπ½) β§ π΅ β (πΌβπΎ) β§ πΆ β (πΌβπΏ))) | ||
Theorem | 3wlkdlem7 29419 | Lemma 7 for 3wlkd 29423. (Contributed by AV, 7-Feb-2021.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) β β’ (π β (π½ β V β§ πΎ β V β§ πΏ β V)) | ||
Theorem | 3wlkdlem8 29420 | Lemma 8 for 3wlkd 29423. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) β β’ (π β ((πΉβ0) = π½ β§ (πΉβ1) = πΎ β§ (πΉβ2) = πΏ)) | ||
Theorem | 3wlkdlem9 29421 | Lemma 9 for 3wlkd 29423. (Contributed by AV, 7-Feb-2021.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) β β’ (π β ({π΄, π΅} β (πΌβ(πΉβ0)) β§ {π΅, πΆ} β (πΌβ(πΉβ1)) β§ {πΆ, π·} β (πΌβ(πΉβ2)))) | ||
Theorem | 3wlkdlem10 29422* | Lemma 10 for 3wlkd 29423. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) β β’ (π β βπ β (0..^(β―βπΉ)){(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ))) | ||
Theorem | 3wlkd 29423 | Construction of a walk from two given edges in a graph. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) & β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (π β πΉ(WalksβπΊ)π) | ||
Theorem | 3wlkond 29424 | 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.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) & β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (π β πΉ(π΄(WalksOnβπΊ)π·)π) | ||
Theorem | 3trld 29425 | 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.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) & β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β (π½ β πΎ β§ π½ β πΏ β§ πΎ β πΏ)) β β’ (π β πΉ(TrailsβπΊ)π) | ||
Theorem | 3trlond 29426 | 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.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) & β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β (π½ β πΎ β§ π½ β πΏ β§ πΎ β πΏ)) β β’ (π β πΉ(π΄(TrailsOnβπΊ)π·)π) | ||
Theorem | 3pthd 29427 | 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.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) & β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β (π½ β πΎ β§ π½ β πΏ β§ πΎ β πΏ)) β β’ (π β πΉ(PathsβπΊ)π) | ||
Theorem | 3pthond 29428 | 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.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) & β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β (π½ β πΎ β§ π½ β πΏ β§ πΎ β πΏ)) β β’ (π β πΉ(π΄(PathsOnβπΊ)π·)π) | ||
Theorem | 3spthd 29429 | 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.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) & β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β (π½ β πΎ β§ π½ β πΏ β§ πΎ β πΏ)) & β’ (π β π΄ β π·) β β’ (π β πΉ(SPathsβπΊ)π) | ||
Theorem | 3spthond 29430 | 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.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) & β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β (π½ β πΎ β§ π½ β πΏ β§ πΎ β πΏ)) & β’ (π β π΄ β π·) β β’ (π β πΉ(π΄(SPathsOnβπΊ)π·)π) | ||
Theorem | 3cycld 29431 | 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.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) & β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β (π½ β πΎ β§ π½ β πΏ β§ πΎ β πΏ)) & β’ (π β π΄ = π·) β β’ (π β πΉ(CyclesβπΊ)π) | ||
Theorem | 3cyclpd 29432 | 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.) |
β’ π = β¨βπ΄π΅πΆπ·ββ© & β’ πΉ = β¨βπ½πΎπΏββ© & β’ (π β ((π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π))) & β’ (π β ((π΄ β π΅ β§ π΄ β πΆ) β§ (π΅ β πΆ β§ π΅ β π·) β§ πΆ β π·)) & β’ (π β ({π΄, π΅} β (πΌβπ½) β§ {π΅, πΆ} β (πΌβπΎ) β§ {πΆ, π·} β (πΌβπΏ))) & β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β (π½ β πΎ β§ π½ β πΏ β§ πΎ β πΏ)) & β’ (π β π΄ = π·) β β’ (π β (πΉ(CyclesβπΊ)π β§ (β―βπΉ) = 3 β§ (πβ0) = π΄)) | ||
Theorem | upgr3v3e3cycl 29433* | 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.) |
β’ πΈ = (EdgβπΊ) & β’ π = (VtxβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(CyclesβπΊ)π β§ (β―βπΉ) = 3) β βπ β π βπ β π βπ β π (({π, π} β πΈ β§ {π, π} β πΈ β§ {π, π} β πΈ) β§ (π β π β§ π β π β§ π β π))) | ||
Theorem | uhgr3cyclexlem 29434 | Lemma for uhgr3cyclex 29435. (Contributed by AV, 12-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ ((((π΄ β π β§ π΅ β π) β§ π΄ β π΅) β§ ((π½ β dom πΌ β§ {π΅, πΆ} = (πΌβπ½)) β§ (πΎ β dom πΌ β§ {πΆ, π΄} = (πΌβπΎ)))) β π½ β πΎ) | ||
Theorem | uhgr3cyclex 29435* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ ((πΊ β UHGraph β§ ((π΄ β π β§ π΅ β π β§ πΆ β π) β§ (π΄ β π΅ β§ π΄ β πΆ β§ π΅ β πΆ)) β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 3 β§ (πβ0) = π΄)) | ||
Theorem | umgr3cyclex 29436* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ ((πΊ β UMGraph β§ (π΄ β π β§ π΅ β π β§ πΆ β π) β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 3 β§ (πβ0) = π΄)) | ||
Theorem | umgr3v3e3cycl 29437* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ (πΊ β UMGraph β (βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 3) β βπ β π βπ β π βπ β π ({π, π} β πΈ β§ {π, π} β πΈ β§ {π, π} β πΈ))) | ||
Theorem | upgr4cycl4dv4e 29438* | 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.) |
β’ π = (VtxβπΊ) & β’ πΈ = (EdgβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(CyclesβπΊ)π β§ (β―βπΉ) = 4) β βπ β π βπ β π βπ β π βπ β π ((({π, π} β πΈ β§ {π, π} β πΈ) β§ ({π, π} β πΈ β§ {π, π} β πΈ)) β§ ((π β π β§ π β π β§ π β π) β§ (π β π β§ π β π β§ π β π)))) | ||
Syntax | cconngr 29439 | Extend class notation with connected graphs. |
class ConnGraph | ||
Definition | df-conngr 29440* | 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 29382 and dfconngr1 29441. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ ConnGraph = {π β£ [(Vtxβπ) / π£]βπ β π£ βπ β π£ βπβπ π(π(PathsOnβπ)π)π} | ||
Theorem | dfconngr1 29441* | 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.) |
β’ ConnGraph = {π β£ [(Vtxβπ) / π£]βπ β π£ βπ β (π£ β {π})βπβπ π(π(PathsOnβπ)π)π} | ||
Theorem | isconngr 29442* | The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ π = (VtxβπΊ) β β’ (πΊ β π β (πΊ β ConnGraph β βπ β π βπ β π βπβπ π(π(PathsOnβπΊ)π)π)) | ||
Theorem | isconngr1 29443* | The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ π = (VtxβπΊ) β β’ (πΊ β π β (πΊ β ConnGraph β βπ β π βπ β (π β {π})βπβπ π(π(PathsOnβπΊ)π)π)) | ||
Theorem | cusconngr 29444 | A complete hypergraph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ ((πΊ β UHGraph β§ πΊ β ComplGraph) β πΊ β ConnGraph) | ||
Theorem | 0conngr 29445 | A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ β β ConnGraph | ||
Theorem | 0vconngr 29446 | A graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ ((πΊ β π β§ (VtxβπΊ) = β ) β πΊ β ConnGraph) | ||
Theorem | 1conngr 29447 | A graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.) |
β’ ((πΊ β π β§ (VtxβπΊ) = {π}) β πΊ β ConnGraph) | ||
Theorem | conngrv2edg 29448* | A vertex in a connected graph with more than one vertex is incident with at least one edge. Formerly part of proof for vdgn0frgrv2 29548. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 4-Apr-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β ConnGraph β§ π β π β§ 1 < (β―βπ)) β βπ β ran πΌ π β π) | ||
Theorem | vdn0conngrumgrv2 29449 | 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.) |
β’ π = (VtxβπΊ) β β’ (((πΊ β ConnGraph β§ πΊ β UMGraph) β§ (π β π β§ 1 < (β―βπ))) β ((VtxDegβπΊ)βπ) β 0) | ||
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 29451 resp. iseupth 29454. (EulerPathsβπΊ) is the set of all Eulerian paths in graph πΊ, see eupths 29453. 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 29451, whereas there is no explicit definition for Eulerian circuits (yet): The statement "β¨πΉ, πβ© is an Eulerian circuit" is formally expressed by (πΉ(EulerPathsβπΊ)π β§ πΉ(CircuitsβπΊ)π). Each Eulerian path can be made an Eulerian circuit by adding an edge which connects the endpoints of the Eulerian path (see eupth2eucrct 29470). Vice versa, removing one edge from a graph with an Eulerian circuit results in a graph with an Eulerian path, see eucrct2eupth 29498. An Eulerian path does not have to be a path in the meaning of definition df-pths 28973, 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 29494, it is shown that a pseudograph with an Eulerian path has either zero or two vertices of odd degree, and eulercrct 29495 shows that a pseudograph with an Eulerian circuit has only vertices of even degree. | ||
Syntax | ceupth 29450 | Extend class notation with Eulerian paths. |
class EulerPaths | ||
Definition | df-eupth 29451* | Define the set of all Eulerian paths on an arbitrary graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
β’ EulerPaths = (π β V β¦ {β¨π, πβ© β£ (π(Trailsβπ)π β§ π:(0..^(β―βπ))βontoβdom (iEdgβπ))}) | ||
Theorem | releupth 29452 | The set (EulerPathsβπΊ) of all Eulerian paths on πΊ 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.) |
β’ Rel (EulerPathsβπΊ) | ||
Theorem | eupths 29453* | The Eulerian paths on the graph πΊ. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ (EulerPathsβπΊ) = {β¨π, πβ© β£ (π(TrailsβπΊ)π β§ π:(0..^(β―βπ))βontoβdom πΌ)} | ||
Theorem | iseupth 29454 | The property "β¨πΉ, πβ© is an Eulerian path on the graph πΊ". An Eulerian path is defined as bijection πΉ from the edges to a set 0...(π β 1) and a function π:(0...π)βΆπ into the vertices such that for each 0 β€ π < π, πΉ(π) is an edge from π(π) to π(π + 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.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(EulerPathsβπΊ)π β (πΉ(TrailsβπΊ)π β§ πΉ:(0..^(β―βπΉ))βontoβdom πΌ)) | ||
Theorem | iseupthf1o 29455 | The property "β¨πΉ, πβ© is an Eulerian path on the graph πΊ". An Eulerian path is defined as bijection πΉ from the edges to a set 0...(π β 1) and a function π:(0...π)βΆπ into the vertices such that for each 0 β€ π < π, πΉ(π) is an edge from π(π) to π(π + 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.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(EulerPathsβπΊ)π β (πΉ(WalksβπΊ)π β§ πΉ:(0..^(β―βπΉ))β1-1-ontoβdom πΌ)) | ||
Theorem | eupthi 29456 | 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.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(EulerPathsβπΊ)π β (πΉ(WalksβπΊ)π β§ πΉ:(0..^(β―βπΉ))β1-1-ontoβdom πΌ)) | ||
Theorem | eupthf1o 29457 | The πΉ 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.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(EulerPathsβπΊ)π β πΉ:(0..^(β―βπΉ))β1-1-ontoβdom πΌ) | ||
Theorem | eupthfi 29458 | 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.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(EulerPathsβπΊ)π β dom πΌ β Fin) | ||
Theorem | eupthseg 29459 | The π-th edge in an eulerian path is the edge having π(π) and π(π + 1) as endpoints . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ ((πΉ(EulerPathsβπΊ)π β§ π β (0..^(β―βπΉ))) β {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ))) | ||
Theorem | upgriseupth 29460* | The property "β¨πΉ, πβ© is an Eulerian path on the pseudograph πΊ". (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.) |
β’ πΌ = (iEdgβπΊ) & β’ π = (VtxβπΊ) β β’ (πΊ β UPGraph β (πΉ(EulerPathsβπΊ)π β (πΉ:(0..^(β―βπΉ))β1-1-ontoβdom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}))) | ||
Theorem | upgreupthi 29461* | 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.) |
β’ πΌ = (iEdgβπΊ) & β’ π = (VtxβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β (πΉ:(0..^(β―βπΉ))β1-1-ontoβdom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))})) | ||
Theorem | upgreupthseg 29462 | The π-th edge in an eulerian path is the edge from π(π) to π(π + 1). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π β§ π β (0..^(β―βπΉ))) β (πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}) | ||
Theorem | eupthcl 29463 | An Eulerian path has length β―(πΉ), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
β’ (πΉ(EulerPathsβπΊ)π β (β―βπΉ) β β0) | ||
Theorem | eupthistrl 29464 | An Eulerian path is a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.) (Revised by AV, 18-Feb-2021.) |
β’ (πΉ(EulerPathsβπΊ)π β πΉ(TrailsβπΊ)π) | ||
Theorem | eupthiswlk 29465 | An Eulerian path is a walk. (Contributed by AV, 6-Apr-2021.) |
β’ (πΉ(EulerPathsβπΊ)π β πΉ(WalksβπΊ)π) | ||
Theorem | eupthpf 29466 | The π 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.) |
β’ (πΉ(EulerPathsβπΊ)π β π:(0...(β―βπΉ))βΆ(VtxβπΊ)) | ||
Theorem | eupth0 29467 | 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ ((π΄ β π β§ πΌ = β ) β β (EulerPathsβπΊ){β¨0, π΄β©}) | ||
Theorem | eupthres 29468 | The restriction β¨π», πβ© of an Eulerian path β¨πΉ, πβ© to an initial segment of the path (of length π) forms an Eulerian path on the subgraph π 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.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ π» = (πΉ prefix π) & β’ π = (π βΎ (0...π)) & β’ (Vtxβπ) = π β β’ (π β π»(EulerPathsβπ)π) | ||
Theorem | eupthp1 29469 | Append one path segment to an Eulerian path β¨πΉ, πβ© to become an Eulerian path β¨π», πβ© of the supergraph π obtained by adding the new edge to the graph πΊ. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 7-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) (Revised by AV, 8-Apr-2024.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β πΌ β Fin) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β Β¬ π΅ β dom πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ π = (β―βπΉ) & β’ (π β πΈ β (EdgβπΊ)) & β’ (π β {(πβπ), πΆ} β πΈ) & β’ (iEdgβπ) = (πΌ βͺ {β¨π΅, πΈβ©}) & β’ π» = (πΉ βͺ {β¨π, π΅β©}) & β’ π = (π βͺ {β¨(π + 1), πΆβ©}) & β’ (Vtxβπ) = π & β’ ((π β§ πΆ = (πβπ)) β πΈ = {πΆ}) β β’ (π β π»(EulerPathsβπ)π) | ||
Theorem | eupth2eucrct 29470 | Append one path segment to an Eulerian path β¨πΉ, πβ© which may not be an (Eulerian) circuit to become an Eulerian circuit β¨π», πβ© of the supergraph π obtained by adding the new edge to the graph πΊ. (Contributed by AV, 11-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) (Revised by AV, 8-Apr-2024.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β πΌ β Fin) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β Β¬ π΅ β dom πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ π = (β―βπΉ) & β’ (π β πΈ β (EdgβπΊ)) & β’ (π β {(πβπ), πΆ} β πΈ) & β’ (iEdgβπ) = (πΌ βͺ {β¨π΅, πΈβ©}) & β’ π» = (πΉ βͺ {β¨π, π΅β©}) & β’ π = (π βͺ {β¨(π + 1), πΆβ©}) & β’ (Vtxβπ) = π & β’ ((π β§ πΆ = (πβπ)) β πΈ = {πΆ}) & β’ (π β πΆ = (πβ0)) β β’ (π β (π»(EulerPathsβπ)π β§ π»(Circuitsβπ)π)) | ||
Theorem | eupth2lem1 29471 | Lemma for eupth2 29492. (Contributed by Mario Carneiro, 8-Apr-2015.) |
β’ (π β π β (π β if(π΄ = π΅, β , {π΄, π΅}) β (π΄ β π΅ β§ (π = π΄ β¨ π = π΅)))) | ||
Theorem | eupth2lem2 29472 | Lemma for eupth2 29492. (Contributed by Mario Carneiro, 8-Apr-2015.) |
β’ π΅ β V β β’ ((π΅ β πΆ β§ π΅ = π) β (Β¬ π β if(π΄ = π΅, β , {π΄, π΅}) β π β if(π΄ = πΆ, β , {π΄, πΆ}))) | ||
Theorem | trlsegvdeglem1 29473 | Lemma for trlsegvdeg 29480. (Contributed by AV, 20-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) β β’ (π β ((πβπ) β π β§ (πβ(π + 1)) β π)) | ||
Theorem | trlsegvdeglem2 29474 | Lemma for trlsegvdeg 29480. (Contributed by AV, 20-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β Fun (iEdgβπ)) | ||
Theorem | trlsegvdeglem3 29475 | Lemma for trlsegvdeg 29480. (Contributed by AV, 20-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β Fun (iEdgβπ)) | ||
Theorem | trlsegvdeglem4 29476 | Lemma for trlsegvdeg 29480. (Contributed by AV, 21-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β dom (iEdgβπ) = ((πΉ β (0..^π)) β© dom πΌ)) | ||
Theorem | trlsegvdeglem5 29477 | Lemma for trlsegvdeg 29480. (Contributed by AV, 21-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β dom (iEdgβπ) = {(πΉβπ)}) | ||
Theorem | trlsegvdeglem6 29478 | Lemma for trlsegvdeg 29480. (Contributed by AV, 21-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β dom (iEdgβπ) β Fin) | ||
Theorem | trlsegvdeglem7 29479 | Lemma for trlsegvdeg 29480. (Contributed by AV, 21-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β dom (iEdgβπ) β Fin) | ||
Theorem | trlsegvdeg 29480 | Formerly part of proof of eupth2lem3 29489: If a trail in a graph πΊ induces a subgraph π with the vertices π of πΊ and the edges being the edges of the walk, and a subgraph π with the vertices π of πΊ and the edges being the edges of the walk except the last one, and a subgraph π with the vertices π of πΊ and one edges being the last edge of the walk, then the vertex degree of any vertex π of πΊ within π is the sum of the vertex degree of π within π and the vertex degree of π within π. 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β ((VtxDegβπ)βπ) = (((VtxDegβπ)βπ) + ((VtxDegβπ)βπ))) | ||
Theorem | eupth2lem3lem1 29481 | Lemma for eupth2lem3 29489. (Contributed by AV, 21-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β ((VtxDegβπ)βπ) β β0) | ||
Theorem | eupth2lem3lem2 29482 | Lemma for eupth2lem3 29489. (Contributed by AV, 21-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) β β’ (π β ((VtxDegβπ)βπ) β β0) | ||
Theorem | eupth2lem3lem3 29483* | Lemma for eupth2lem3 29489, formerly part of proof of eupth2lem3 29489: If a loop {(πβπ), (πβ(π + 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) & β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπ)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) & β’ (π β if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ)))) β β’ ((π β§ (πβπ) = (πβ(π + 1))) β (Β¬ 2 β₯ (((VtxDegβπ)βπ) + ((VtxDegβπ)βπ)) β π β if((πβ0) = (πβ(π + 1)), β , {(πβ0), (πβ(π + 1))}))) | ||
Theorem | eupth2lem3lem4 29484* | Lemma for eupth2lem3 29489, formerly part of proof of eupth2lem3 29489: 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) & β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπ)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) & β’ (π β if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ)))) & β’ (π β (πΌβ(πΉβπ)) β π« π) β β’ ((π β§ (πβπ) β (πβ(π + 1)) β§ (π = (πβπ) β¨ π = (πβ(π + 1)))) β (Β¬ 2 β₯ (((VtxDegβπ)βπ) + ((VtxDegβπ)βπ)) β π β if((πβ0) = (πβ(π + 1)), β , {(πβ0), (πβ(π + 1))}))) | ||
Theorem | eupth2lem3lem5 29485* | Lemma for eupth2 29492. (Contributed by AV, 25-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) & β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπ)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) & β’ (π β (πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}) β β’ (π β (πΌβ(πΉβπ)) β π« π) | ||
Theorem | eupth2lem3lem6 29486* | Formerly part of proof of eupth2lem3 29489: 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 (πβ0) and (πβπ): 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) & β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπ)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) & β’ (π β (πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}) β β’ ((π β§ (πβπ) β (πβ(π + 1)) β§ (π β (πβπ) β§ π β (πβ(π + 1)))) β (Β¬ 2 β₯ (((VtxDegβπ)βπ) + ((VtxDegβπ)βπ)) β π β if((πβ0) = (πβ(π + 1)), β , {(πβ0), (πβ(π + 1))}))) | ||
Theorem | eupth2lem3lem7 29487* | Lemma for eupth2lem3 29489: Combining trlsegvdeg 29480, eupth2lem3lem3 29483, eupth2lem3lem4 29484 and eupth2lem3lem6 29486. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 27-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β Fun πΌ) & β’ (π β π β (0..^(β―βπΉ))) & β’ (π β π β π) & β’ (π β πΉ(TrailsβπΊ)π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (Vtxβπ) = π) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) & β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπ)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) & β’ (π β (πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}) β β’ (π β (Β¬ 2 β₯ ((VtxDegβπ)βπ) β π β if((πβ0) = (πβ(π + 1)), β , {(πβ0), (πβ(π + 1))}))) | ||
Theorem | eupthvdres 29488 | Formerly part of proof of eupth2 29492: 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΊ β π) & β’ (π β Fun πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ π» = β¨π, (πΌ βΎ (πΉ β (0..^(β―βπΉ))))β© β β’ (π β (VtxDegβπ») = (VtxDegβπΊ)) | ||
Theorem | eupth2lem3 29489* | Lemma for eupth2 29492. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΊ β UPGraph) & β’ (π β Fun πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ π» = β¨π, (πΌ βΎ (πΉ β (0..^π)))β© & β’ π = β¨π, (πΌ βΎ (πΉ β (0..^(π + 1))))β© & β’ (π β π β β0) & β’ (π β (π + 1) β€ (β―βπΉ)) & β’ (π β π β π) & β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπ»)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) β β’ (π β (Β¬ 2 β₯ ((VtxDegβπ)βπ) β π β if((πβ0) = (πβ(π + 1)), β , {(πβ0), (πβ(π + 1))}))) | ||
Theorem | eupth2lemb 29490* | Lemma for eupth2 29492 (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 29492. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΊ β UPGraph) & β’ (π β Fun πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) β β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegββ¨π, (πΌ βΎ (πΉ β (0..^0)))β©)βπ₯)} = β ) | ||
Theorem | eupth2lems 29491* | Lemma for eupth2 29492 (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 29492. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΊ β UPGraph) & β’ (π β Fun πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) β β’ ((π β§ π β β0) β ((π β€ (β―βπΉ) β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegββ¨π, (πΌ βΎ (πΉ β (0..^π)))β©)βπ₯)} = if((πβ0) = (πβπ), β , {(πβ0), (πβπ)})) β ((π + 1) β€ (β―βπΉ) β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegββ¨π, (πΌ βΎ (πΉ β (0..^(π + 1))))β©)βπ₯)} = if((πβ0) = (πβ(π + 1)), β , {(πβ0), (πβ(π + 1))})))) | ||
Theorem | eupth2 29492* | 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.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΊ β UPGraph) & β’ (π β Fun πΌ) & β’ (π β πΉ(EulerPathsβπΊ)π) β β’ (π β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)} = if((πβ0) = (πβ(β―βπΉ)), β , {(πβ0), (πβ(β―βπΉ))})) | ||
Theorem | eulerpathpr 29493* | 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.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β (β―β{π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)}) β {0, 2}) | ||
Theorem | eulerpath 29494* | 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.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β UPGraph β§ (EulerPathsβπΊ) β β ) β (β―β{π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)}) β {0, 2}) | ||
Theorem | eulercrct 29495* | A pseudograph with an Eulerian circuit β¨πΉ, πβ© (an "Eulerian pseudograph") has only vertices of even degree. (Contributed by AV, 12-Mar-2021.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π β§ πΉ(CircuitsβπΊ)π) β βπ₯ β π 2 β₯ ((VtxDegβπΊ)βπ₯)) | ||
Theorem | eucrctshift 29496* | Cyclically shifting the indices of an Eulerian circuit β¨πΉ, πβ© results in an Eulerian circuit β¨π», πβ©. (Contributed by AV, 15-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΉ(CircuitsβπΊ)π) & β’ π = (β―βπΉ) & β’ (π β π β (0..^π)) & β’ π» = (πΉ cyclShift π) & β’ π = (π₯ β (0...π) β¦ if(π₯ β€ (π β π), (πβ(π₯ + π)), (πβ((π₯ + π) β π)))) & β’ (π β πΉ(EulerPathsβπΊ)π) β β’ (π β (π»(EulerPathsβπΊ)π β§ π»(CircuitsβπΊ)π)) | ||
Theorem | eucrct2eupth1 29497 | Removing one edge (πΌβ(πΉβπ)) from a nonempty graph πΊ with an Eulerian circuit β¨πΉ, πβ© results in a graph π with an Eulerian path β¨π», πβ©. This is the special case of eucrct2eupth 29498 (with π½ = (π β 1)) where the last segment/edge of the circuit is removed. (Contributed by AV, 11-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ (π β πΉ(CircuitsβπΊ)π) & β’ (Vtxβπ) = π & β’ (π β 0 < (β―βπΉ)) & β’ (π β π = ((β―βπΉ) β 1)) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) & β’ π» = (πΉ prefix π) & β’ π = (π βΎ (0...π)) β β’ (π β π»(EulerPathsβπ)π) | ||
Theorem | eucrct2eupth 29498* | Removing one edge (πΌβ(πΉβπ½)) from a graph πΊ with an Eulerian circuit β¨πΉ, πβ© results in a graph π with an Eulerian path β¨π», πβ©. (Contributed by AV, 17-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ (π β πΉ(EulerPathsβπΊ)π) & β’ (π β πΉ(CircuitsβπΊ)π) & β’ (Vtxβπ) = π & β’ (π β π = (β―βπΉ)) & β’ (π β π½ β (0..^π)) & β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β ((0..^π) β {π½})))) & β’ πΎ = (π½ + 1) & β’ π» = ((πΉ cyclShift πΎ) prefix (π β 1)) & β’ π = (π₯ β (0..^π) β¦ if(π₯ β€ (π β πΎ), (πβ(π₯ + πΎ)), (πβ((π₯ + πΎ) β π)))) β β’ (π β π»(EulerPathsβπ)π) | ||
According to Wikipedia ("Seven Bridges of KΓΆnigsberg", 9-Mar-2021, https://en.wikipedia.org/wiki/Seven_Bridges_of_Koenigsberg): "The Seven Bridges of KΓΆnigsberg 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 KΓΆnigsberg 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 KΓΆnigsberg 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 KΓΆnigsberg graph πΊ is a multigraph consisting of 4 vertices and 7 edges, represented by the following ordered pair: πΊ = β¨(0...3), β¨β{0, 1}{0, 2} {0, 3}{1, 2}{1, 2}{2, 3}{2, 3}ββ©β©, see konigsbergumgr 29504. konigsberg 29510 shows that the KΓΆnigsberg graph has no Eulerian path, thus the KΓΆnigsberg Bridge problem has no solution. | ||
Theorem | konigsbergvtx 29499 | The set of vertices of the KΓΆnigsberg graph πΊ. (Contributed by AV, 28-Feb-2021.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ (VtxβπΊ) = (0...3) | ||
Theorem | konigsbergiedg 29500 | The indexed edges of the KΓΆnigsberg graph πΊ. (Contributed by AV, 28-Feb-2021.) |
β’ π = (0...3) & β’ πΈ = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© & β’ πΊ = β¨π, πΈβ© β β’ (iEdgβπΊ) = β¨β{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}ββ© |
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