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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 Fulkerson–Chen–Anstee theorem ： ウィキペディア英語版 Fulkerson–Chen–Anstee theorem ---- The Fulkerson–Chen–Anstee theorem is a result in graph theory, a branch of combinatorics. It provides one of two known approaches solving the digraph realization problem, i.e. it gives a necessary and sufficient condition for pairs of nonnegative integers $((a\_1,b\_1),\backslash ldots,(a\_n,b\_n))$ to be the indegree-outdegree pairs of a simple directed graph; a sequence obeying these conditions is called "digraphic". D. R. Fulkerson 〔D.R. Fulkerson: ''Zero-one matrices with zero trace.'' In: ''Pacific J. Math.'' No. 12, 1960, pp. 831–836〕 (1960) obtained a characterization analogous to the classical Erdős–Gallai theorem for graphs, but in contrast to this solution with exponentially many inequalities. In 1966 Chen 〔Wai-Kai Chen: ''On the realization of a (''p'',''s'')-digraph with prescribed degrees .'' In: ''Journal of the Franklin Institute'' No. 6, 1966, pp. 406–422〕 improved this result in demanding the additional constraint that the integer pairs must be sorted in non-increasing lexicographical order leading to n inequalities. Anstee 〔Richard Anstee: ''Properties of a class of (0,1)-matrices covering a given matrix.'' In: ''Can. J. Math.'', 1982, pp. 438–453〕 (1982) observed in a different context that it is suffient to have $a\_1\backslash geq\backslash cdots\backslash geq\; a\_n$. Berger 〔Annabell Berger: ''A Note on the Characterization of Digraphic Sequences'' In: ''Discrete Mathematics'', 2014, pp. 38–41〕 reinvented this result and gives a direct proof. ==Theorem statement== A sequence $((a\_1,b\_1\; ),\backslash ldots,(a\_n,b\_n))$ of nonnegative integer pairs with $a\_1\backslash geq\backslash cdots\backslash geq\; a\_n$ is digraphic if and only if $\backslash sum\_^a\_i=\backslash sum\_^b\_i$ and the following inequality holds for ''k'' such that $1\; \backslash leq\; k\; \backslash leq\; n$: : $$ \sum^k_ a_i\leq \sum^k_ \min(b_i,k-1)+ \sum^n_ \min(b_i,k) 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「Fulkerson–Chen–Anstee theorem」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.