diff --git a/chapter19.tex b/chapter19.tex index 1163cec..d000a7e 100644 --- a/chapter19.tex +++ b/chapter19.tex @@ -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. diff --git a/chapter20.tex b/chapter20.tex index 9ca3eb0..1e7a8d0 100644 --- a/chapter20.tex +++ b/chapter20.tex @@ -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. diff --git a/list.tex b/list.tex index b7f34d7..be0c952 100644 --- a/list.tex +++ b/list.tex @@ -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}, Addison–Wesley, 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.