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Some fixes

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Antti H S Laaksonen 2017-02-26 11:29:50 +02:00
parent 9f5bef4b7e
commit ee009bd9dd
3 changed files with 31 additions and 28 deletions
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13
chapter19.tex
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@ -398,8 +398,9 @@ so we have successfully constructed an Eulerian circuit.
\index{Hamiltonian path} \index{Hamiltonian path}
A \key{Hamiltonian path}\footnote{ A \key{Hamiltonian path}
W. R. Hamilton (1805--1865) was an Irish mathematician.} is a path %\footnote{W. R. Hamilton (1805--1865) was an Irish mathematician.}
is a path
that visits each node in the graph exactly once. that visits each node in the graph exactly once.
For example, the graph For example, the graph
\begin{center} \begin{center}
@ -485,12 +486,12 @@ Also stronger results have been achieved:
\begin{itemize} \begin{itemize}
\item \item
\index{Dirac's theorem} \index{Dirac's theorem}
\key{Dirac's theorem} \cite{dir52}: \key{Dirac's theorem}: %\cite{dir52}
If the degree of each node is at least $n/2$, If the degree of each node is at least $n/2$,
the graph contains a Hamiltonian path. the graph contains a Hamiltonian path.
\item \item
\index{Ore's theorem} \index{Ore's theorem}
\key{Ore's theorem} \cite{ore60}: \key{Ore's theorem}: %\cite{ore60}
If the sum of degrees of each non-adjacent pair of nodes If the sum of degrees of each non-adjacent pair of nodes
is at least $n$, is at least $n$,
the graph contains a Hamiltonian path. the graph contains a Hamiltonian path.
@ -529,7 +530,9 @@ It is possible to implement this solution in $O(2^n n^2)$ time.
\index{De Bruijn sequence} \index{De Bruijn sequence}
A \key{De Bruijn sequence}\footnote{N. G. de Bruijn (1918--2012) was a Dutch mathematician.} is a string that contains A \key{De Bruijn sequence}
%\footnote{N. G. de Bruijn (1918--2012) was a Dutch mathematician.}
is a string that contains
every string of length $n$ every string of length $n$
exactly once as a substring, for a fixed exactly once as a substring, for a fixed
alphabet of $k$ characters. alphabet of $k$ characters.

6
chapter20.tex
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@ -930,7 +930,7 @@ The maximum flow of this graph is as follows:
\index{Hall's theorem} \index{Hall's theorem}
\index{perfect matching} \index{perfect matching}
\key{Hall's theorem} \cite{hal35} can be used to find out \key{Hall's theorem} can be used to find out
whether a bipartite graph has a matching whether a bipartite graph has a matching
that contains all left or right nodes. that contains all left or right nodes.
If the number of left and right nodes is the same, If the number of left and right nodes is the same,
@ -1020,7 +1020,7 @@ has at least one endpoint in the set.
In a general graph, finding a minimum node cover In a general graph, finding a minimum node cover
is a NP-hard problem. is a NP-hard problem.
However, if the graph is bipartite, However, if the graph is bipartite,
\key{Kőnig's theorem} \cite{kon31} tells us that \key{Kőnig's theorem} tells us that
the size of a minimum node cover the size of a minimum node cover
and the size of a maximum matching are always equal. and the size of a maximum matching are always equal.
Thus, we can calculate the size of a minimum node cover Thus, we can calculate the size of a minimum node cover
@ -1409,7 +1409,7 @@ An \key{antichain} is a set of nodes of a graph
such that there is no path such that there is no path
from any node to another node from any node to another node
using the edges of the graph. using the edges of the graph.
\key{Dilworth's theorem} \cite{dil50} states that \key{Dilworth's theorem} states that
in a directed acyclic graph, the size of in a directed acyclic graph, the size of
a minimum general path cover a minimum general path cover
equals the size of a maximum antichain. equals the size of a maximum antichain.

40
list.tex
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@ -59,15 +59,15 @@
A note on two problems in connexion with graphs. A note on two problems in connexion with graphs.
\emph{Numerische Mathematik}, 1(1):269--271, 1959. \emph{Numerische Mathematik}, 1(1):269--271, 1959.
\bibitem{dil50} % \bibitem{dil50}
R. P. Dilworth. % R. P. Dilworth.
A decomposition theorem for partially ordered sets. % A decomposition theorem for partially ordered sets.
\emph{Annals of Mathematics}, 51(1):161--166, 1950. % \emph{Annals of Mathematics}, 51(1):161--166, 1950.
\bibitem{dir52} % \bibitem{dir52}
G. A. Dirac. % G. A. Dirac.
Some theorems on abstract graphs. % Some theorems on abstract graphs.
\emph{Proceedings of the London Mathematical Society}, 3(1):69--81, 1952. % \emph{Proceedings of the London Mathematical Society}, 3(1):69--81, 1952.
\bibitem{edm65} \bibitem{edm65}
J. Edmonds. J. Edmonds.
@ -147,10 +147,10 @@
Computer Science and Computational Biology}, Computer Science and Computational Biology},
Cambridge University Press, 1997. Cambridge University Press, 1997.
\bibitem{hal35} % \bibitem{hal35}
P. Hall. % P. Hall.
On representatives of subsets. % On representatives of subsets.
\emph{Journal London Mathematical Society} 10(1):26--30, 1935. % \emph{Journal London Mathematical Society} 10(1):26--30, 1935.
On representatives of subsets. J. London Math. Soc, 10(1), 26-30. On representatives of subsets. J. London Math. Soc, 10(1), 26-30.
@ -211,10 +211,10 @@
D. E. Knuth. D. E. Knuth.
\emph{The Art of Computer Programming. Volume 3: Sorting and Searching}, Addison–Wesley, 1998 (2nd edition). \emph{The Art of Computer Programming. Volume 3: Sorting and Searching}, Addison–Wesley, 1998 (2nd edition).
\bibitem{kon31} % \bibitem{kon31}
D. Kőnig. % D. Kőnig.
Gráfok és mátrixok. % Gráfok és mátrixok.
\emph{Matematikai és Fizikai Lapok}, 38(1):116--119, 1931. % \emph{Matematikai és Fizikai Lapok}, 38(1):116--119, 1931.
\bibitem{kru56} \bibitem{kru56}
J. B. Kruskal. J. B. Kruskal.
@ -231,10 +231,10 @@
An $O(n \log n)$ algorithm for finding all repetitions in a string. An $O(n \log n)$ algorithm for finding all repetitions in a string.
\emph{Journal of Algorithms}, 5(3):422--432, 1984. \emph{Journal of Algorithms}, 5(3):422--432, 1984.
\bibitem{ore60} % \bibitem{ore60}
Ø. Ore. % Ø. Ore.
Note on Hamilton circuits. % Note on Hamilton circuits.
\emph{The American Mathematical Monthly}, 67(1):55, 1960. % \emph{The American Mathematical Monthly}, 67(1):55, 1960.
\bibitem{pac13} \bibitem{pac13}
J. Pachocki and J. Radoszweski. J. Pachocki and J. Radoszweski.