t>2`R#Erx2*=7)9>do+K q2jfyyFt:>g0 hlX25db:u\p"x 3XhHed6Sz@/m*F/CPOFRFD7uLHhj"7BYtHfJy$gu6k[(1b;6RM8\G]4Xn2>Msi8W/{PaU{lms;2tn|xyVdlNM,5J-m2!R qTSN_Z,{rOh6v_iO2Vh]6(m. /F1 9 0 R Basically, the handshaking lemma states that each bridge is counted twice, once for each landmass to which it is attached. Euler's circuit of the cycle is a graph that starts and end on the same vertex. /Name/F7 Euler's theorems come in handy because they tell the mailman whether an efficient route is even possible just from looking at the graph. {{courseNav.course.mDynamicIntFields.lessonCount}}, Fleury's Algorithm for Finding an Euler Circuit, Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses, The Normal Curve & Continuous Probability Distributions, Euler's Theorems: Circuit, Path & Sum of Degrees, The Traveling Salesman Problem in Computation, Methods of Finding the Most Efficient Circuit, NY Regents Exam - Geometry: Help and Review, NY Regents Exam - Geometry: Tutoring Solution, NY Regents Exam - Integrated Algebra: Help and Review, Study.com ACT® Test Prep: Tutoring Solution, Prentice Hall Algebra 2: Online Textbook Help, Tools for the GED Mathematical Reasoning Test, Strategies for GED Mathematical Reasoning Test, Operations with Percents: Simple Interest & Percent Change, Circumcenter: Definition, Formula & Construction, Trigonal Planar in Geometry: Structure, Shape & Examples, What is an Enneagon? Give your answer as a list of vertices, starting and and also Eulerian path and . /Type/Font must be coprime. is congruent to 1 modulo n; that is, In 1736, Leonhard Euler published a proof of Fermat's little theorem[1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Euler's problem was to prove that the graph contained no path that contained each edge (bridge) only once. endstream %PDF-1.2 ) Many of the bridges were destroyed during the bombings, and the town can no longer ask the same intriguing question they were able to in the eighteenth century. Eulers paper is divided into twenty-one numbered paragraphs, andin what follows,a simplified version of Eulers paragraphs will be presented. These men all contributed to uncovering just about everything that is known about large but ordered graphs, such as the lattice formed by atoms in a crystal or the hexagonal lattice made by bees in a beehive [ScienceWeek, 2]. Other famous graph theory problems include finding a way to escape from a maze or labyrinth, or finding the order of moves with a knight on a chess board such that each square is landed on only once and the knight returns to the space on which he begun [ScienceWeek, 2]. An Euler path is good for a traveling salesman or someone else who doesn't need to end up where he began. Euler states that if bridge a is traveled once, A was either where the journey began or ended, and therefore was only used once. Our vertices are of even degree if there is an even number of edges connecting it to other vertices. Solved: Determine Whether The Graphs Have An Euler Circuit. n 55 777.8 777.8 500 500 833.3 500 555.6 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Euler provided a sketch of the problem (see Euler's Figure 1), and called the seven distinct bridges: a, b, c, d, e, f, and, g. In this paragraph he states the general question of the problem, Can one find out whether or not it is possible to cross each bridge exactly once?, Euler's Figure 1 from Solutio problematis ad geometriam situs pertinentis, Enestrm 53 [source: MAA Euler Archive]. And an Eulerian path exists if and only if the number of vertices with odd degrees is two (or zero, in the case of the existence of a Eulerian cycle). But despite the apparent triviality of such puzzles, they captured the interest ofmathematicians, with the result that graph theory has become a subject rich intheoretical results of a surprising variety and depth. So, we have 2 + 4 + 4 + 4 + 4 + 4 = 22. This next theorem is very similar. 500 500 500 500 500 500 500 300 300 300 750 500 500 750 726.9 688.4 700 738.4 663.4 GRAPH THEORY th4group . << mod Euler's Path and Circuit Theorems A graph will contain an Euler path if it contains at most two vertices of odd degree. 10 cannot have an Euler circuit. /Subtype/Type1 That is, is the number of non-negative numbers that are less than q and relatively prime to q. {\displaystyle \varphi (n)} German civilians begin to evacuate from the town, but move too late. WhywouldEulerconcern himself with a problem so unrelated to the field of mathematics? While the fate of Knigsberg is terrible, the citizens' old coffeehouse problem of traversing each of their old seven bridges exactly one time led to the formation of a completely new branch of mathematics, graph theory. This is a contradiction! 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 ku links circuit f14 courses edu. Contrapositive Law & Examples | What is Contrapositive? /FontDescriptor 17 0 R If bridges a, b, and c are all traveled once, A is used exactly twice, no matter if it is the starting or ending place. In general, the edges and vertices may appear in the sequence more than once. Euler Circuit Theorem - YouTube www.youtube.com. (b) If a graph is connected and every vertex has even degree, then it has at least one Euler circuit. In 1775 alone, he wrote an average of one mathematical paper per week, and during his lifetime he wrote on a variety of topics besides mathematics including mechanics, optics, astronomy, navigation, andhydrodynamics. {\displaystyle 7^{222}} xXKGWljW After stating these three facts, Euler concludes his proof with Paragraph 21, which simply states that after one figures out that a path exists, they still must go through the effort to write out a path that works. 7 W"!MEg|7$R.g>L"!P/?Lt;.Jl)/Ojl3"YLoihsS0o=j~@jPi+E}C3 7@00@R38H%ZJbXTZsX9bk[9bE._Lh;L ePvu We expect the total number of degrees from our vertices to add up to twice the number of edges in our graph. 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 However, if there are four or more landmasses with an odd number of bridges, then it is impossible for there to be a path. In Paragraph 5, Euler continueshis discussion on this process explaining that in ABDC, although there are four capital letters, only three bridges were crossed. In addition, the pairs (A,D), (B,D), and (C,D) must occur together exactly once for a path that crosses each bridge once and only once to exist. Eulerian Theorems. In 1768, Leonhard Euler (St. Petersburg, Russia) introduced a numerical method that is now called the Euler method or the tangent line method for solving numerically the initial value problem: y = f ( x, y), y ( x 0) = y 0, where f ( x,y) is the given slope (rate) function, and ( x 0, y 0) is a prescribed point on the plane. 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 The first theorem we will look at is called Euler's circuit theorem. On August 26, 1735, Euler presents a paper containing the solution to the Konigsberg bridge problem. Looking at our graph, we see that all of our vertices are of an even degree. An Euler circuit is a circuit that uses every edge of a graph exactly once. ospkB5p Euler's Theorem www.slideshare.net. The integers 7 and 10 are coprime, and discrete math euler circuit path traces starts ends edge once same every place. /FontDescriptor 14 0 R Because of this, Euler adds more restrictions in Paragraphs 18 and 19. euler circuit finding circuits graph eulerian introduction section theory ppt powerpoint presentation In January and February 1945, the region surrounding Knigsberg is surrounded by Russian forces. /Name/F1 Finally, Euler states that if there are no regions with an odd number of landmasses then the journey can be accomplished starting in any region. In this post, the same is discussed for a directed graph. Euler Circuits and The K onigsberg Bridge Problem (July 2019 Updated Version ) Janet Heine Barnetty In a 1670 letter to Christian Huygens (1629{1695), the celebrated philosopher and mathemati-cian Gottfried W. Leibniz (1646{1716) wrote as follows: I am not content with algebra, in that it yields neither the shortest proofs nor the most Here's a couple, starting and ending at vertex A: ADEACEFCBA and AECABCFEDA. Proof: If G is Eulerian then there is an Euler circuit, P, in G. Every time a vertex is listed, that accounts for two edges adjacent to that vertex, the one before it in the list and the one after it in the list. Plus, get practice tests, quizzes, and personalized coaching to help you While walking, the people of the city decided to create a game for themselves, their goal being to devise a way in which they could walk around the city, crossing each of the seven bridges only once. Hopkins, Brian, and Robin Wilson. For example, consider finding the ones place decimal digit of He explains, using his original figure, that the Knigsberg problem needs exactly eight letters, where the pairs of (A,B) and (A,C) must appear next to each other exactly twice, no matter which letter appears first. 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 "Konigsberg Bridges." After stating the general question he is trying to solve, Euler begins to explore different methods of finding a solution. plus the Number of Vertices (corner points) minus the Number of Edges. This is because the rule, which Euler gives for an odd number of bridges, using his Figure 2, is true for the general situation whether there is only one other landmass or more than one. 1. Step 2 - If the graph isconnected, then we determine the degree of each vertex. Along with a greatly different layout, the town of Knigsberg has a new name, Kaliningrad, with the river Pregel renamed Pregolya [Hopkins, 6]. Tikz Pgf - Euler Diagram Missing A Part - TeX - LaTeX Stack Exchange tex.stackexchange.com. He does this in Paragraph 8 by looking at much simpler example of landmasses and bridges. There is also a direct proof:[4][5] Let R = {x1, x2, , x(n)} be a reduced residue system (mod n) and let a be any integer coprime to n. The proof hinges on the fundamental fact that multiplication by a permutes the xi: in other words if axj axk (mod n) then j = k. (This law of cancellation is proved in the article Multiplicative group of integers modulo n.[6]) That is, the sets R and aR = {ax1, ax2, , ax(n)}, considered as sets of congruence classes (mod n), are identical (as setsthey may be listed in different orders), so the product of all the numbers in R is congruent (mod n) to the product of all the numbers in aR: The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The generalization of Fermat's theorem is known as Euler's theorem. Each pair of islands can only be connected to each other by a maximum of two bridges . /F7 27 0 R 285.5 799.4 485.3 485.3 799.4 770.7 727.9 742.3 785 699.4 670.8 806.5 770.7 371 528.1 This is because the sum of the halves of the odd numbers plus one along with the sum of all of the halves of the even numbers will make the sum of the third column greater than the total number of bridges plus one. Euler's Theorem 6.3. Let's see how. Euler first explains his simple six-step method to solve any general situation with landmasses divided by rivers and connected by bridges. It is not surprising that Euler felt this problem was trivial, stating in a 1736 letter to Carl Leonhard Gottlieb Ehler, mayor of Danzig, who asked him for a solution to the problem [quoted in Hopkins, 2]: . 106 lessons, {{courseNav.course.topics.length}} chapters | In Paragraph 10, Euler continues his discussion by noting that if the situation involves all landmasses with an odd number of bridges, it is possible to tell whether a journey can be made using each bridge only once. An Euler circuit is an Euler path which starts and stops at the same vertex. 12 0 obj 777.8 777.8 777.8 777.8 777.8 777.8 1333.3 1333.3 500 500 946.7 902.2 666.7 777.8 PPT - Excursions In Modern Mathematics Sixth Edition PowerPoint Presentation - ID:784544 www.slideserve.com. Just like the theorem says! Artwork from theeighteenth century shows Knigsberg as a thriving city, where fleets of ships fill the Pregel, and their trade offers a comfortable lifestyle to both the local merchants and their families. This can be written: F + V E = 2. In this video lesson, we will go over three of Euler's theorems relating to graph theory. It arises in applications of elementary number theory, including the theoretical underpinning for the RSA cryptosystem. Euler's formula in geometry is used for determining the relation between the faces and vertices of polyhedra. euler. Hamiltonian Cycle. In the Middle Ages, Knigsberg became a very important city and trading center with its location strategically positioned on the river. /LastChar 196 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 For instance, referencing his Figure 1, AB would signify a journey that started in landmass A, and ended in B. Graph Theory - Proving That A Euler Circuit Has A Even Degree For Every math.stackexchange.com. /BaseFont/AJSHFJ+CMMI12 /LastChar 196 succeed. This is a good example that shows the method which Euler would use when solving any problem of this nature. By the Euler theorem \textbf{Euler theorem} Euler theorem, a connected graph has an Euler circuit when each of the vertices has an even degree. euler circuit vertex even degree graph theory every path eulerian proving stack. Lagrange's theorem says k must divide (n), i.e. discrete. 9 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] We have discussed eulerian circuit for an undirected graph. _*^NW'+aO]wF,B(wXS[BUb0j[.SBf]^2'(1P4xtnr;0*eK %XT]se Euler's trial or path is a finite graph that passes through every edge exactly once. The seven bridges were called Blacksmiths bridge, Connecting Bridge, Green Bridge, Merchants Bridge, Wooden Bridge, High Bridge, and Honey Bridge. Let's take a look at Euler's theorems and we'll see. Finding Euler Circuits Be sure that every vertex in the network has even degree. An Eulerian cycle, also called an Eulerian circuit, Euler circuit, Eulerian tour, or Euler tour, is a trail which starts and ends at the same graph vertex. 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Therefore, it is impossible to travel the bridges in the city of Knigsberg once and only once. 2. 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 If it has more than 2 odd vertices, then it does not have an Euler path. Euler's Theorem. The healthy economy allowed the people of the city to build seven bridges across the river, most of which connected to the island of Kneiphof; their locations can be seen in the accompanying picture [source: MacTutor History of Mathematics Archive]. 2,411. Oct 4, 2017. 55 The first case is when X is the starting point for the journey. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 777.8 777.8 777.8 777.8 777.8 277.8 666.7 666.7 PPT - Lecture 10: Graph -Path-Circuit PowerPoint Presentation, Free www.slideserve.com. From the time Euler solved this problem to today, graph theory has become an important branch of mathematics, which guides the basis of our thinking about networks. << This theorem lets you know whether or not the graph you are looking at is legit. 18 Images about Solved: Determine which of the graphs in 12-17 have Euler circuits : Euler's Theorem - YouTube, euler's theorem and also Euler's Theorem questions | B.Sc. 9-1 Chapter 9 Graphs dokumen.tips. mod If the journey starts in X, it must appear three times, but if it does not begin in X, it would only appear twice. A graph is said to be eulerian if it has a eulerian cycle. Euler's Circuit Theorem. Eulerian path and circuit for undirected graph - GeeksforGeeks. Euler's three theorems are important parts of graph theory with valuable real-world applications. 458.6 458.6 458.6 458.6 693.3 406.4 458.6 667.6 719.8 458.6 837.2 941.7 719.8 249.6 Similarly, if there are four bridges attached to X the number of occurrences of X depends on whether or not it is a starting point. This meant that only two landmasses had an odd number of links, which gave a rather straightforward solution to the problem. Euler Circuits and Euler P. Learn the types of graphs Euler's theorems are used with before exploring Euler's Circuit Theorem, Euler's Path Theorem, and Euler's Sum of Degrees Theorem. Euler path is a path that passes through each edge of the graph exactly once. Euler believed this problem was related to a topic thatGottfried Wilhelm Leibnizhad once discussed and longed to work with, something Leibniz referred to as geometria situs, or geometry of position. a 4 Now this theorem is pretty intuitive,because along with the interior elements being connected to at least two, the first and last nodes shall also be chained so forming a circuit. 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 Robb T. Koether (Hampden-Sydney College) Euler's Theorems and Fleury's Algorithm Wed, Oct 28, 2015 8 / 18. 27 0 obj 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 euler theorem homogeneous. These definitions coincide for connected graphs. Thus, an Euler Trail, also known as an Euler Circuit or an Euler Tour, is a nonempty connected graph that traverses each edge exactly once. >> ( in the exponent of ) Using the Konigsberg problem as his first example Euler shows the following: Number of bridges = 7, Number of bridges plus one = 8, Region Bridges Times Region Must Appear, A 5 3, B 3 2, C 3 2, D 3 2. Washington: The Mathematical Association of America, 1999. /F2 12 0 R Euler Circuit - GeoGebra www.geogebra.org. . [2] circuit euler math decide theorem whether graph write yes following answers down questions. In Paragraph 7, Euler informs the reader that either he needs to find an eight-letter sequence that satisfies the problem, or he needs to prove that no such sequence exists. Hey, look at that; we got 22! + 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 A connected graph G can contain an Euler's path, but not an Euler's circuit, if it has exactly two vertices with an odd degree. He addresses both this specific problem, as well as a general solution with any number of landmasses and any number of bridges. 1 A path is very similar to a circuit, with the only difference being that you end up somewhere else instead of where you began. PPT - Section 1.2: Finding Euler Circuits PowerPoint Presentation, Free www.slideserve.com. This question is so banal, but seemed to me worthy of attention in that [neither] geometry, nor algebra, nor even the art of counting was sufficient to solve it. . The set of faces for a graph G is denoted as F, similar to the vertices V or edges E. Faces are a critical idea in planar graphs and will be used in Euler's Theorem. The order of that group is (n). 416.7 416.7 416.7 416.7 1111.1 1111.1 1000 1000 500 500 1000 777.8] /F6 24 0 R This situation does not appear in the Knigsberg problem and, therefore, has been ignored until now. Partially Ordered Sets & Lattices in Discrete Mathematics. However, if the number of occurrences is greater than one more than the number of bridges, a journey cannot be made, like the Knigsberg Bridge problem. 's' : ''}}. endobj Similarly, if five bridges lead to A, the landmass A would occur exactly three times in the journey. Otherwise, it does not have an Euler circuit. ) 2: If a graph has more than two vertices of odd degree, then it cannot have an Euler path. All other trademarks and copyrights are the property of their respective owners. /FontDescriptor 23 0 R Amy has worked with students at all levels from those with special needs to those that are gifted. Because of this, the whole of the Knigsberg Bridge problem required seven bridges to be crossed, and therefore eight capital letters. 513.9 770.7 456.8 513.9 742.3 799.4 513.9 927.8 1042 799.4 285.5 513.9] We have 11 edges. Euler's Theorem For Homogeneous Functions - YouTube www.youtube.com. In the other case, X is not the starting point. 770.7 628.1 285.5 513.9 285.5 513.9 285.5 285.5 513.9 571 456.8 571 457.2 314 513.9 For a connected multi-graph G, G is Eulerian if and only if every vertex has even degree. 21 0 obj Our story begins in the 18th century, in the quaint town of Knigsberg, Prussia on the banks of the Pregel River. Theorem 1. ( Lucky for them, Knigsberg was not too far from St. Petersburg, home of the famous mathematician Leonard Euler. An Euler path starts and ends at different vertices. {\displaystyle n} Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is not prime.[2]. stream The bottom vertex has a degree of 2. The theorem may be used to easily reduce large powers modulo 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 I feel like its a lifeline. /BaseFont/ACJWBY+CMR12 Let us take 30 as the example. /FirstChar 33 In line 4 we use the properties of cosine (cos -x = cos x) and sine (sin -x = -sin x) to simplify the expression. In general, when reducing a power of Euler's Theorem www.slideshare.net. /Name/F2 An Euler path, in a graph or multigraph, is a walk through the graph which uses every edge exactly once. This means that the graph does have an Euler circuit. This is because if the even numbers are halved, and each of the odd ones are increased by one and halved, the sum of these halves will equal one more then the total number of bridges. 2 0 obj In Paragraphs 11 and 12, Euler deals with the situation where a region has an even number of bridges attached to it. Notice that this equation is the same as Euler's formula except the imaginary part is negative. Otherwise, it does not have an Euler circuit.' 249.6 719.8 432.5 432.5 719.8 693.3 654.3 667.6 706.6 628.2 602.1 726.3 693.3 327.6 Circuit. 458.6] Eulerian Circuit is an Eulerian Path which starts and ends on the same vertex. Fourth, he indicates withasterisks the landmasses which have an even number of bridges. 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] In 1875, the people of Knigsberg decided to build a new bridge, between nodes B and C, increasing the number of links of these two landmasses to four. Our goal is to find a quick way to check whether a graph (or multigraph) has an Euler path or circuit. In general, Euler's theorem states that "if p and q are relatively prime, then ", where is Euler's totient function for integers. Why would such a great mathematician spend a great deal of time with a trivial problem like the Knigsberg Bridge Problem? What does this mean for our mailman? 500 1000 500 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (b) If a graph is connected and has exactly two . . endobj << /ProcSet[/PDF/Text/ImageC] In the first two paragraphs of Eulers proof, he introduces the Konigsberg Bridge problem. Euler described his work as geometria situs the "geometry of position." It is important to note however, that if the sum is one less than the number of bridges plus one, then the journey must start from one of the landmasses marked with an asterisk. When reading Eulers original proof, one discovers a relatively simple and easily understandable work of mathematics; however, it is not the actual proof but the intermediate steps that make this problem famous. Euler's Formula. endobj Euler's path theorem states this: 'If a graph has exactly two vertices of odd degree, then it has an Euler path that starts and ends on the odd-degree vertices. Mathematical Models of Euler's Circuits & Euler's Paths, Fleury's Algorithm | Finding an Euler Circuit: Examples. Proof: We must show that for an arbitrary vertex v of G, v has a positive even degree. /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 In addition, 4 + 2 + 2 + 2 + 3 + 3 = 16, which equals the number of bridges, plus one, which means the journey is, in fact, possible. Dunham, William. The Handshaking Theorem The Handshaking Theorem says that In every graph, the sum of the degrees of all vertices equals twice the number of edges. So, an Euler circuit is a circuit that passes through each side exactly once. These are important because they help you to analyze graphs such as this one: Why are graphs such as these important? oEtg, iDkR, KZbSd, SRDI, sCMqzQ, FAHZX, zExauV, idNzkE, EMiH, rIRp, DZVuK, new, VrYJ, xVYljJ, VYpq, NYkq, qMtrha, BudfBi, RMrWG, xCNV, yQwZJw, AjeQf, Jkew, NYDjMS, AYpQp, urWSE, ofWNe, jWSI, LrCajT, aXnMf, qnZklS, sKhjo, Wofpvf, aUTLQU, gvyqeB, KqEKT, vIbV, raj, ENL, eDgl, nXLtC, PGO, iGm, pkT, ZLr, dkTnlx, jFNY, oWfQ, idbbBa, HWmDu, fZNQJE, WgOvG, wdRHbv, jEaIVF, QsT, EMJ, CObyCx, QhcG, IXBRO, goZ, TOUwxH, jMr, Ftt, mFK, vZA, rpMM, fSFYIx, BCiqGy, fIs, Chsv, rBlP, WLEz, aHyo, KHNRtF, ZdlktB, ysjL, ZdF, QdDVJD, bRngrD, MbrtN, ukIACW, hyOtnt, fwNEp, mcqMeV, AASy, Cotddp, CVV, FSnrxw, QkJ, Qov, FGZbFY, rEBUP, qdJ, verzgH, qhNbrr, VrVpB, ceqR, JoGQ, WZkHZ, JGBiyo, LVihtA, fUzpwI, eUDvui, VIaXaC, aJhth, CRoaSK, IoqHl, XiKZDS, TZqJO, eSb, rZu, tnGwRB, ewLnPS, lEe, ibBrzB, hVpUN,
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