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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Discrepancy of hypergraphs ： ウィキペディア英語版 Discrepancy of hypergraphs Discrepancy of hypergraphs is an area of discrepancy theory. == Hypergraph discrepancies in two colors == In the classical setting, we aim at partitioning the vertices of a hypergraph into two classes in such a way that ideally each hyperedge contains the same number of vertices in both classes. A partition into two classes can be represented by a coloring $\backslash chi\; \backslash rightarrow\; \backslash$. We call -1 and +1 ''colors''. The color-classes $\backslash chi^(-1)$ and $\backslash chi^(+1)$ form the corresponding partition. For a hyperedge $E\; \backslash in\; \backslash mathcal$, set :$\backslash chi(E)\; :=\; \backslash sum\_\; \backslash chi(v).$ The ''discrepancy of $\backslash mathcal$ with respect to $\backslash chi$'' and the ''discrepancy of $\backslash mathcal$'' are defined by :$disc(\backslash mathcal,\backslash chi)\; :=\; \backslash max\_)\; :=\; \backslash min\_,\; \backslash chi).$ These notions as well as the term 'discrepancy' seem to have appeared for the first time in a paper of Beck.〔J. Beck: "Roth's estimate of the discrepancy of integer sequences is nearly sharp", page 319-325. Combinatorica, 1, 1981〕 Earlier results on this problem include the famous lower bound on the discrepancy of arithmetic progressions by Roth〔K. F. Roth: "Remark concerning integer sequences", pages 257–260. Acta Arithmetica 9, 1964〕 and upper bounds for this problem and other results by Erdős and Spencer〔J. Spencer: "A remark on coloring integers", pages 43–44. Canad. Math. Bull. 15, 1972.〕〔P. Erdős and J. Spencer: "Imbalances in k-colorations", pages 379–385. Networks 1, 1972.〕 and Sárközi (described on p. 39).〔P. Erdős and J. Spencer: "Probabilistic Methods in Combinatorics." Akadémia Kiadó, Budapest, 1974.〕 At that time, discrepancy problems were called quasi-Ramsey problems. To get some intuition for this concept, let's have a look at a few examples. * If all edges of $\backslash mathcal$ intersect trivially, i.e. $E\_1\; \backslash cap\; E\_2\; =\; \backslash emptyset$ for any two distinct edges $E\_1,\; E\_2\; \backslash in\; \backslash mathcal$, then the discrepancy is zero, if all edges have even cardinality, and one, if there is an odd cardinality edge. * The other extreme is marked by the ''complete hypergraph'' $(V,\; 2^V)$. In this case the discrepancy is $\backslash lceil\; \backslash frac\; |V|\backslash rceil$. Any 2-coloring will have a color class of at least this size, and this set is also an edge. On the other hand, any coloring $\backslash chi$ with color classes of size $\backslash lceil\; \backslash frac\; |V|\backslash rceil$ and $\backslash lfloor\; \backslash frac\; |V|\backslash rfloor$ proves that the discrepancy is not larger than $\backslash lceil\; \backslash frac\; |V|\backslash rceil$. It seems that the discrepancy reflects how chaotic the hyperedges of $\backslash mathcal$ intersect. Things are not that easy, however, as the following example shows. * Set $n=4k$, $k\; \backslash in\; \backslash mathcal$ and $\backslash mathcal\_n\; =\; ((),\; \backslash )$. Now $\backslash mathcal\_n$ has many (more than $\backslash binom^2\; =\; \backslash Theta(\backslash frac\; 1\; n\; 2^n)$) complicatedly intersecting edges, but discrepancy zero. The last example shows that we cannot expect to determine the discrepancy by looking at a single parameter like the number of hyperedges. Still, the size of the hypergraph yields first upper bounds. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Discrepancy of hypergraphs」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.