diff --git a/chapter19.tex b/chapter19.tex index 7654741..e3276b5 100644 --- a/chapter19.tex +++ b/chapter19.tex @@ -22,7 +22,9 @@ problem and no efficient algorithm is known for solving the problem. \index{Eulerian path} -An \key{Eulerian path} is a path +An \key{Eulerian path}\footnote{L. Euler (1707--1783) studied such paths in 1736 +when he solved the famous Königsberg bridge problem. +This was the birth of graph theory.} is a path that goes exactly once through each edge in the graph. For example, the graph \begin{center} @@ -222,7 +224,8 @@ from node 2 to node 5: \index{Hierholzer's algorithm} -\key{Hierholzer's algorithm} is an efficient +\key{Hierholzer's algorithm}\footnote{The algorithm was published +in 1873 after Hierholzer's death \cite{hie73}.} is an efficient method for constructing an Eulerian circuit. The algorithm consists of several rounds, @@ -395,7 +398,8 @@ so we have successfully constructed an Eulerian circuit. \index{Hamiltonian path} -A \key{Hamiltonian path} 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} @@ -481,12 +485,12 @@ Also stronger results have been achieved: \begin{itemize} \item \index{Dirac's theorem} -\key{Dirac's theorem}: +\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}: +\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. @@ -525,7 +529,7 @@ It is possible to implement this solution in $O(2^n n^2)$ time. \index{De Bruijn sequence} -A \key{De Bruijn sequence} 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. @@ -546,7 +550,7 @@ and each edge adds one character to the string. The following graph corresponds to the above example: \begin{center} -\begin{tikzpicture} +\begin{tikzpicture}[scale=0.8] \node[draw, circle] (00) at (-3,0) {00}; \node[draw, circle] (11) at (3,0) {11}; \node[draw, circle] (01) at (0,2) {01}; @@ -633,7 +637,8 @@ a complete tour will be found quickly. \index{heuristic} \index{Warnsdorff's rule} -\key{Warnsdorff's rule} is a simple and effective heuristic +\key{Warnsdorff's rule}\footnote{This heuristic was proposed +in Warnsdorff's book \cite{war23} in 1823.} is a simple and effective heuristic for finding a knight's tour. Using the rule, it is possible to efficiently construct a tour even on a large board. diff --git a/list.tex b/list.tex index d55e6b4..a0d925c 100644 --- a/list.tex +++ b/list.tex @@ -44,6 +44,11 @@ A note on two problems in connexion with graphs. \emph{Numerische Mathematik}, 1(1):269--271, 1959. +\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. Paths, trees, and flowers. @@ -117,6 +122,11 @@ A dynamic programming approach to sequencing problems. \emph{Journal of the Society for Industrial and Applied Mathematics}, 10(1):196--210, 1962. +\bibitem{hie73} + C. Hierholzer and C. Wiener. + Über die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. + \emph{Mathematische Annalen}, 6(1), 30--32, 1873. + \bibitem{hop71} J. E. Hopcroft and J. D. Ullman. A linear list merging algorithm. @@ -169,6 +179,11 @@ 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{pac13} J. Pachocki and J. Radoszweski. Where to use and how not to use polynomial string hashing. @@ -213,4 +228,9 @@ Dimer problem in statistical mechanics -- an exact result. \emph{Philosophical Magazine}, 6(68):1061--1063, 1961. +\bibitem{war23} + H. C. von Warnsdorff. + \emph{Des Rösselsprungs einfachste und allgemeinste Lösung}. + Schmalkalden, 1823. + \end{thebibliography} \ No newline at end of file