Some fixes
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@ -398,8 +398,9 @@ so we have successfully constructed an Eulerian circuit.
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\index{Hamiltonian path}
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A \key{Hamiltonian path}\footnote{
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W. R. Hamilton (1805--1865) was an Irish mathematician.} is a path
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A \key{Hamiltonian path}
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%\footnote{W. R. Hamilton (1805--1865) was an Irish mathematician.}
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is a path
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that visits each node in the graph exactly once.
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For example, the graph
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\begin{center}
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@ -485,12 +486,12 @@ Also stronger results have been achieved:
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\begin{itemize}
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\item
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\index{Dirac's theorem}
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\key{Dirac's theorem} \cite{dir52}:
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\key{Dirac's theorem}: %\cite{dir52}
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If the degree of each node is at least $n/2$,
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the graph contains a Hamiltonian path.
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\item
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\index{Ore's theorem}
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\key{Ore's theorem} \cite{ore60}:
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\key{Ore's theorem}: %\cite{ore60}
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If the sum of degrees of each non-adjacent pair of nodes
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is at least $n$,
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the graph contains a Hamiltonian path.
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@ -529,7 +530,9 @@ It is possible to implement this solution in $O(2^n n^2)$ time.
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\index{De Bruijn sequence}
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A \key{De Bruijn sequence}\footnote{N. G. de Bruijn (1918--2012) was a Dutch mathematician.} is a string that contains
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A \key{De Bruijn sequence}
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%\footnote{N. G. de Bruijn (1918--2012) was a Dutch mathematician.}
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is a string that contains
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every string of length $n$
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exactly once as a substring, for a fixed
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alphabet of $k$ characters.
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@ -930,7 +930,7 @@ The maximum flow of this graph is as follows:
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\index{Hall's theorem}
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\index{perfect matching}
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\key{Hall's theorem} \cite{hal35} can be used to find out
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\key{Hall's theorem} can be used to find out
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whether a bipartite graph has a matching
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that contains all left or right nodes.
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If the number of left and right nodes is the same,
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@ -1020,7 +1020,7 @@ has at least one endpoint in the set.
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In a general graph, finding a minimum node cover
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is a NP-hard problem.
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However, if the graph is bipartite,
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\key{Kőnig's theorem} \cite{kon31} tells us that
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\key{Kőnig's theorem} tells us that
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the size of a minimum node cover
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and the size of a maximum matching are always equal.
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Thus, we can calculate the size of a minimum node cover
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@ -1409,7 +1409,7 @@ An \key{antichain} is a set of nodes of a graph
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such that there is no path
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from any node to another node
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using the edges of the graph.
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\key{Dilworth's theorem} \cite{dil50} states that
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\key{Dilworth's theorem} states that
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in a directed acyclic graph, the size of
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a minimum general path cover
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equals the size of a maximum antichain.
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40
list.tex
40
list.tex
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@ -59,15 +59,15 @@
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A note on two problems in connexion with graphs.
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\emph{Numerische Mathematik}, 1(1):269--271, 1959.
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\bibitem{dil50}
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R. P. Dilworth.
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A decomposition theorem for partially ordered sets.
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\emph{Annals of Mathematics}, 51(1):161--166, 1950.
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% \bibitem{dil50}
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% R. P. Dilworth.
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% A decomposition theorem for partially ordered sets.
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% \emph{Annals of Mathematics}, 51(1):161--166, 1950.
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\bibitem{dir52}
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G. A. Dirac.
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Some theorems on abstract graphs.
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\emph{Proceedings of the London Mathematical Society}, 3(1):69--81, 1952.
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% \bibitem{dir52}
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% G. A. Dirac.
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% Some theorems on abstract graphs.
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% \emph{Proceedings of the London Mathematical Society}, 3(1):69--81, 1952.
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\bibitem{edm65}
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J. Edmonds.
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@ -147,10 +147,10 @@
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Computer Science and Computational Biology},
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Cambridge University Press, 1997.
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\bibitem{hal35}
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P. Hall.
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On representatives of subsets.
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\emph{Journal London Mathematical Society} 10(1):26--30, 1935.
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% \bibitem{hal35}
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% P. Hall.
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% On representatives of subsets.
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% \emph{Journal London Mathematical Society} 10(1):26--30, 1935.
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On representatives of subsets. J. London Math. Soc, 10(1), 26-30.
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@ -211,10 +211,10 @@
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D. E. Knuth.
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\emph{The Art of Computer Programming. Volume 3: Sorting and Searching}, Addison–Wesley, 1998 (2nd edition).
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\bibitem{kon31}
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D. Kőnig.
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Gráfok és mátrixok.
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\emph{Matematikai és Fizikai Lapok}, 38(1):116--119, 1931.
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% \bibitem{kon31}
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% D. Kőnig.
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% Gráfok és mátrixok.
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% \emph{Matematikai és Fizikai Lapok}, 38(1):116--119, 1931.
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\bibitem{kru56}
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J. B. Kruskal.
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@ -231,10 +231,10 @@
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An $O(n \log n)$ algorithm for finding all repetitions in a string.
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\emph{Journal of Algorithms}, 5(3):422--432, 1984.
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\bibitem{ore60}
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Ø. Ore.
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Note on Hamilton circuits.
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\emph{The American Mathematical Monthly}, 67(1):55, 1960.
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% \bibitem{ore60}
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% Ø. Ore.
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% Note on Hamilton circuits.
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% \emph{The American Mathematical Monthly}, 67(1):55, 1960.
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\bibitem{pac13}
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J. Pachocki and J. Radoszweski.
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