diff --git a/chapter20.tex b/chapter20.tex index 1e7a8d0..9ca3eb0 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} can be used to find out +\key{Hall's theorem} \cite{hal35} 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} tells us that +\key{Kőnig's theorem} \cite{kon31} 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} states that +\key{Dilworth's theorem} \cite{dil50} 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 da165ec..0958ad3 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{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. @@ -117,6 +122,13 @@ 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. + + On representatives of subsets. J. London Math. Soc, 10(1), 26-30. + \bibitem{hel62} M. Held and R. M. Karp. A dynamic programming approach to sequencing problems. @@ -164,6 +176,11 @@ 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{kru56} J. B. Kruskal. On the shortest spanning subtree of a graph and the traveling salesman problem.