Some fixes

This commit is contained in:
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

View File

@ -398,8 +398,9 @@ so we have successfully constructed an Eulerian circuit.
\index{Hamiltonian path}
A \key{Hamiltonian path}\footnote{
W. R. Hamilton (1805--1865) was an Irish mathematician.} is a path
A \key{Hamiltonian path}
%\footnote{W. R. Hamilton (1805--1865) was an Irish mathematician.}
is a path
that visits each node in the graph exactly once.
For example, the graph
\begin{center}
@ -485,12 +486,12 @@ Also stronger results have been achieved:
\begin{itemize}
\item
\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 $n/2$,
the graph contains a Hamiltonian path.
\item
\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
is at least $n$,
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}
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 $n$
exactly once as a substring, for a fixed
alphabet of $k$ characters.

View File

@ -930,7 +930,7 @@ The maximum flow of this graph is as follows:
\index{Hall's theorem}
\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
that contains all left or right nodes.
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
is a NP-hard problem.
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
and the size of a maximum matching are always equal.
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
from any node to another node
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
a minimum general path cover
equals the size of a maximum antichain.

View File

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