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| [[File:konigsburg graph.svg|thumb|165px|The [[Seven Bridges of Königsberg|Königsberg Bridges]] graph. This graph is not Eulerian, therefore, a solution does not exist.]]
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you liked this short article and you would like to acquire extra data regarding [http://www.youtube.com/watch?v=90z1mmiwNS8 Dentists in DC] kindly visit the page. |
| [[File:Labelled Eulergraph.svg|thumb|Every vertex of this graph has an even [[degree (graph theory)|degree]], therefore this is an Eulerian graph. Following the edges in alphabetical order gives an Eulerian circuit/cycle.]]
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| In [[graph theory]], an '''Eulerian trail''' (or '''Eulerian path''') is a [[Trail (graph theory)|trail]] in a graph which visits every [[edge (graph theory)|edge]] exactly once. Similarly, an '''Eulerian circuit''' or '''Eulerian cycle''' is an Eulerian trail which starts and ends on the same [[vertex (graph theory)|vertex]]. They were first discussed by [[Leonhard Euler]] while solving the famous [[Seven Bridges of Königsberg]] problem in 1736. Mathematically the problem can be stated like this:
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| :Given the graph on the right, is it possible to construct a path (or a [[cycle (graph theory)|cycle]], i.e. a path starting and ending on the same vertex) which visits each edge exactly once?
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| Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even [[degree (graph theory)|degree]], and stated without proof that connected graphs with all vertices of even degree have an Eulerian circuit. The first complete proof of this latter claim was published posthumously in 1873 by [[Carl Hierholzer]].<ref>N. L. Biggs, E. K. Lloyd and R. J. Wilson, Graph Theory 1736–1936, Clarendon Press, Oxford, 1976, 8–9, ISBN 0-19-853901-0.</ref>
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| 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.<ref>{{cite journal |doi=10.1137/0128070 |author=C. L. Mallows, N. J. A. Sloane |title=Two-graphs, switching classes and Euler graphs are equal in number |journal=SIAM Journal on Applied Mathematics |volume=28 |year=1975 |pages=876–880 |jstor=2100368 |issue=4 |ref=harv}}</ref> | |
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| For the existence of Eulerian trails it is necessary that no more than two vertices have an odd degree; this means the Königsberg graph is ''not'' Eulerian. If there are no vertices of odd degree, all Eulerian trails are circuits. If there are exactly two vertices of odd degree, all Eulerian trails start at one of them and end at the other. A graph that has an Eulerian trail but not an Eulerian circuit is called '''semi-Eulerian'''.
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| ==Definition==
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| An '''Eulerian trail''',<ref name="pathcycle">Some people reserve the terms ''path'' and ''cycle'' to mean ''non-self-intersecting'' path and cycle. A (potentially) self-intersecting path is known as a '''trail''' or an '''open walk'''; and a (potentially) self-intersecting cycle, a '''circuit''' or a '''closed walk'''. This ambiguity can be avoided by using the terms Eulerian trail and Eulerian circuit when self-intersection is allowed.</ref> or '''Euler walk''' in an [[undirected graph]] is a walk that uses each edge exactly once. If such a walk exists, the graph is called '''traversable''' or '''semi-eulerian'''.<ref>Jun-ichi Yamaguchi, [http://jwilson.coe.uga.edu/EMAT6680/Yamaguchi/emat6690/essay1/GT.html Introduction of Graph Theory].</ref>
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| An '''Eulerian cycle''',<ref name="pathcycle"/> '''Eulerian circuit''' or '''Euler tour''' in an undirected graph is a [[cycle (graph theory)|cycle]] that uses each edge exactly once. If such a cycle exists, the graph is called '''Eulerian''' or '''unicursal'''.<ref>Schaum's outline of theory and problems of graph theory By V. K. Balakrishnan [http://books.google.co.uk/books?id=1NTPbSehvWsC&lpg=PA60&dq=unicursal&pg=PA60#v=onepage&q=unicursal&f=false].</ref> The term "Eulerian graph" is also sometimes used in a weaker sense to denote a graph where every vertex has even degree. For finite [[connected graph]]s the two definitions are equivalent, while a possibly unconnected graph is Eulerian in the weaker sense if and only if each connected component has an Eulerian cycle.
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| For [[directed graph]]s, "path" has to be replaced with ''[[directed path (graph theory)|directed path]]'' and "cycle" with ''[[directed cycle]]''.
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| The definition and properties of Eulerian trails, cycles and graphs are valid for [[multigraph]]s as well.
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| An '''Eulerian orientation''' of an undirected graph ''G'' is an assignment of a direction to each edge of ''G'' such that, at each vertex ''v'', the indegree of ''v'' equals the outdegree of ''v''. Such an orientation exists for any undirected graph in which every vertex has even degree, and may be found by constructing an Euler tour in each connected component of ''G'' and then orienting the edges according to the tour.<ref>{{citation | |
| | last = Schrijver | first = A. | authorlink = Alexander Schrijver
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| | doi = 10.1007/BF02579193
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| | issue = 3-4
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| | journal = Combinatorica
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| | mr = 729790
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| | pages = 375–380
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| | title = Bounds on the number of Eulerian orientations
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| | volume = 3
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| | year = 1983}}.</ref> Every Eulerian orientation of a connected graph is a [[strong orientation]], an orientation that makes the resulting directed graph [[strongly connected]].
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| == Properties ==
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| *An undirected graph has an Eulerian cycle if and only if every vertex has even degree, and all of its vertices with nonzero degree belong to a single [[Connected component (graph theory)|connected component]].
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| *An undirected graph can be decomposed into edge-disjoint [[cycle (graph theory)|cycle]]s if and only if all of its vertices have even degree. So, a graph has an Eulerian cycle if and only if it can be decomposed into edge-disjoint cycles and its nonzero-degree vertices belong to a single connected component.
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| *An undirected graph has an Eulerian trail if and only if at most two vertices have odd degree, and if all of its vertices with nonzero degree belong to a single connected component.
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| *A directed graph has an Eulerian cycle if and only if every vertex has equal [[in degree (graph theory)|in degree]] and [[out degree (graph theory)|out degree]], and all of its vertices with nonzero degree belong to a single [[strongly connected component]]. Equivalently, a directed graph has an Eulerian cycle if and only if it can be decomposed into edge-disjoint [[cycle (graph theory)|directed cycle]]s and all of its vertices with nonzero degree belong to a single strongly connected component.
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| *A directed graph has an Eulerian trail if and only if at most one vertex has ([[out degree (graph theory)|out-degree]]) − ([[in degree (graph theory)|in-degree]]) = 1, at most one vertex has (in-degree) − (out-degree) = 1, every other vertex has equal in-degree and out-degree, and all of its vertices with nonzero degree belong to a single connected component of the underlying undirected graph.
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| == Constructing Eulerian trails and circuits ==
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| === Fleury's algorithm ===
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| '''Fleury's algorithm''' is an elegant but inefficient algorithm which dates to 1883.<ref>{{citation|first=M.|last=Fleury|title=Deux problèmes de Géométrie de situation|language=French|url=http://books.google.com/books?id=l-03AAAAMAAJ&pg=PA257|journal=Journal de mathématiques élémentaires|series=2nd ser.|volume=2|year=1883|pages=257–261}}.</ref> Consider a graph known to have all edges in the same component and at most two vertices of odd degree. The algorithm starts at a vertex of odd degree, or, if the graph has none, it starts with an arbitrarily chosen vertex. At each step it chooses the next edge in the path to be one whose deletion would not disconnect the graph, unless there is no such edge, in which case it picks the remaining edge left at the current vertex. It then moves to the other endpoint of that vertex and deletes the chosen edge. At the end of the algorithm there are no edges left, and the sequence from which the edges were chosen forms an Eulerian cycle if the graph has no vertices of odd degree, or an Eulerian trail if there are exactly two vertices of odd degree.
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| While the ''graph traversal'' in Fleury's algorithm is linear in the number of edges, i.e. ''O''(|''E''|), we also need to factor in the complexity of detecting [[Bridge (graph theory)|bridge]]s. If we are to re-run [[Robert Tarjan|Tarjan]]'s linear time bridge-finding algorithm after the removal of every edge, Fleury's algorithm will have a time complexity of ''O''(|''E''|<sup>2</sup>). A dynamic bridge-finding algorithm of {{harvtxt|Thorup|2000}} allows this to be improved to <math>O(|E|\log^3|E|\log\log|E|)</math> but this is still significantly slower than alternative algorithms.
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| === Hierholzer's algorithm ===
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| [[Carl Hierholzer|Hierholzer]]'s 1873 paper provides a different method for finding Euler cycles that is more efficient than Fleury's algorithm:
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| *Choose any starting vertex ''v'', and follow a trail of edges from that vertex until returning to ''v''. It is not possible to get stuck at any vertex other than ''v'', because the even degree of all vertices ensures that, when the trail enters another vertex ''w'' there must be an unused edge leaving ''w''. The tour formed in this way is a closed tour, but may not cover all the vertices and edges of the initial graph.
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| *As long as there exists a vertex ''v'' that belongs to the current tour but that has adjacent edges not part of the tour, start another trail from ''v'', following unused edges until returning to ''v'', and join the tour formed in this way to the previous tour.
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| By using a data structure such as a [[doubly linked list]] to maintain the set of unused edges incident to each vertex, to maintain the list of vertices on the current tour that have unused edges, and to maintain the tour itself, the individual operations of the algorithm (finding unused edges exiting each vertex, finding a new starting vertex for a tour, and connecting two tours that share a vertex) may be performed in constant time each, so the overall algorithm takes [[linear time]].<ref>{{citation|title=Eulerian Graphs and Related Topics: Part 1, Volume 2|volume=50|series=Annals of Discrete Mathematics|first=Herbert|last=Fleischner|publisher=Elsevier|year=1991|isbn=978-0-444-89110-5|contribution=X.1 Algorithms for Eulerian Trails|pages=X.1–13}}.</ref>
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| ==Counting Eulerian circuits==
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| === Complexity issues ===
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| The number of Eulerian circuits in ''[[Directed graph|digraphs]]'' can be calculated using the so-called '''[[BEST theorem]]''', named after [[N. G. de Bruijn|de '''B'''ruijn]], [[Tatyana Pavlovna Ehrenfest|van Aardenne-'''E'''hrenfest]], [[Cedric Smith (statistician)|'''S'''mith]] and [[W. T. Tutte|'''T'''utte]]. The formula states that the number of Eulerian circuits in a digraph is the product of certain degree factorials and the number of rooted [[Arborescence (graph theory)|arborescences]]. The latter can be computed as a [[determinant]], by the [[matrix tree theorem]], giving a polynomial time algorithm.
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| BEST theorem is first stated in this form in a "note added in proof" to the Aardenne-Ehrenfest and de Bruijn paper (1951). The original proof was [[bijective proof|bijective]] and generalized the [[de Bruijn sequence]]s. It is a variation on an earlier result by Smith and Tutte (1941).
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| Counting the number of Eulerian circuits on ''undirected'' graphs is much more difficult. This problem is known to be [[Sharp-P|#P]]-complete.<ref>Brightwell and [[Peter Winkler|Winkler]], "[http://www.cdam.lse.ac.uk/Reports/Files/cdam-2004-12.pdf Note on Counting Eulerian Circuits]", 2004.</ref> In a positive direction, a [[Markov chain Monte Carlo]] approach, via the ''Kotzig transformations'' (introduced by [[Anton Kotzig]] in 1968) is believed to give a sharp approximation for the number of Eulerian circuits in a graph, though as yet there is no proof of this fact (even for graphs of bounded degree).
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| === Special cases ===
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| The [[Asymptotic analysis|asymptotic formula]] for the number of Eulerian circuits in the [[complete graph]]s was determined by [[Brendan McKay|McKay]] and Robinson (1995):<ref>[[Brendan McKay]] and Robert W. Robinson, [http://cs.anu.edu.au/~bdm/papers/euler.pdf Asymptotic enumeration of eulerian circuits in the complete graph], ''[[Combinatorica]]'', 10 (1995), no. 4, 367–377.</ref>
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| :<math>
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| ec(K_n) = 2^{(n+1)/2}\pi^{1/2} e^{-n^2/2+11/12} n^{(n-2)(n+1)/2} \bigl(1+O(n^{-1/2+\epsilon})\bigr).
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| </math>
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| A similar formula was later obtained by M.I. Isaev (2009) for [[complete bipartite graph]]s:<ref>{{cite journal |author=M.I. Isaev |title=Asymptotic number of Eulerian circuits in complete bipartite graphs |language=Russian |journal=Proc. 52-nd MFTI Conference |year=2009 |place=Moscow |pages=111–114 |ref=harv}}</ref> | |
| :<math>
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| ec(K_{n,n}) = (n/2-1)!^{2n} 2^{n^2-n+1/2}\pi^{-n+1/2} n^{n-1} \bigl(1+O(n^{-1/2+\epsilon})\bigr).
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| </math>
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| == Applications ==
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| Eulerian trails are used in [[bioinformatics]] to reconstruct the [[DNA sequence]] from its fragments.<ref>{{cite journal| last1=Pevzner| first1=Pavel A.| last2=Tang| first2=Haixu| last3=Waterman| first3=Michael S.| year=2001 |title=An Eulerian trail approach to DNA fragment assembly |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=98| pmid=11504945 |issue=17 |pages=9748–9753| pmc=55524 |url=http://www.pnas.org/content/98/17/9748.long |doi=10.1073/pnas.171285098 | bibcode=2001PNAS...98.9748P| ref=harv}}</ref> They are also used in [[CMOS]] circuit design to find an optimal [[logic gate]] ordering.<ref>{{cite journal| last1=Roy| first1=Kuntal| year=2007 |title=Optimum Gate Ordering of CMOS Logic Gates Using Euler Path Approach: Some Insights and Explanations |journal=Journal of Computing and Information Technology |volume=15 |issue=1 |pages=85–92|url=http://cit.srce.unizg.hr/index.php/CIT/article/view/1629/1333 |doi=10.2498/cit.1000731 | ref=harv}}</ref>
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| == See also ==
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| * [[Eulerian matroid]], an abstract generalization of Eulerian graphs
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| * [[Five room puzzle]]
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| *The [[handshaking lemma]], proven by Euler in his original paper, showing that any undirected connected graph has an even number of odd-degree vertices
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| * [[Hamiltonian path]] – a path that visits each ''vertex'' exactly once.
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| * [[Veblen's theorem]], that graphs with even vertex degree can be partitioned into edge-disjoint cycles regardless of their connectivity
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| == Notes ==
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| {{reflist|2}}
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| == References ==
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| * Euler, L., "[http://www.math.dartmouth.edu/~euler/pages/E053.html Solutio problematis ad geometriam situs pertinentis]", ''Comment. Academiae Sci. I. Petropolitanae'' '''8''' (1736), 128–140.
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| *{{citation
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| | last = Hierholzer | first = Carl | author-link = Carl Hierholzer
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| | doi = 10.1007/BF01442866
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| | issue = 1
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| | journal = [[Mathematische Annalen]]
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| | pages = 30–32
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| | title = Ueber die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren
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| | volume = 6
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| | year = 1873}}.
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| * Lucas, E., ''Récréations Mathématiques IV'', Paris, 1921.
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| * Fleury, "Deux problemes de geometrie de situation", ''Journal de mathematiques elementaires'' (1883), 257–261.
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| * [[Tatyana Pavlovna Ehrenfest|T. van Aardenne-Ehrenfest]] and [[Nicolaas Govert de Bruijn|N. G. de Bruijn]], Circuits and trees in oriented linear graphs, ''Simon Stevin'', 28 (1951), 203–217.
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| *{{Citation|first=Mikkel|last=Thorup|authorlink=Mikkel Thorup|year=2000|pages=343–350|doi=10.1145/335305.335345|contribution=Near-optimal {{Sic|hide=y|fully|-}}dynamic graph connectivity|title=[[Symposium on Theory of Computing|Proc. 32nd ACM Symposium on Theory of Computing]]}}
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| * [[W. T. Tutte]] and [[Cedric Smith (statistician)|C. A. B. Smith]], On Unicursal Paths in a Network of Degree 4. ''Amer. Math. Monthly'', 48 (1941), 233–237.
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| == External links ==
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| {{Commons category|Eulerian paths}}
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| * [http://mathforum.org/kb/message.jspa?messageID=3648262&tstart=135 Discussion of early mentions of Fleury's algorithm]
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| [[Category:Graph theory objects]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
If you liked this short article and you would like to acquire extra data regarding Dentists in DC kindly visit the page.