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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 Turán graph ： ウィキペディア英語版 Turán graph The Turán graph ''T''(''n'',''r'') is a complete multipartite graph formed by partitioning a set of ''n'' vertices into ''r'' subsets, with sizes as equal as possible, and connecting two vertices by an edge whenever they belong to different subsets. The graph will have $(n\backslash ,\backslash bmod\backslash ,r)$ subsets of size $\backslash lceil\; n/r\backslash rceil$, and $r-(n\backslash ,\backslash bmod\backslash ,r)$ subsets of size $\backslash lfloor\; n/r\backslash rfloor$. That is, it is a complete ''r''-partite graph :$K\_.$ Each vertex has degree either $n-\backslash lceil\; n/r\backslash rceil$ or $n-\backslash lfloor\; n/r\backslash rfloor$. The number of edges is :$\backslash frac(n^2\; -\; (n\backslash ,\backslash bmod\backslash ,r)\backslash lceil\; n/r\backslash rceil^2\; -\; (r-(n\backslash ,\backslash bmod\backslash ,r))\backslash lfloor\; n/r\backslash rfloor^2)\; \backslash leq\; \backslash left(1-\backslash frac1r\backslash right)\backslash frac.$ It is a regular graph, if ''n'' is divisible by ''r''. ==Turán's theorem== Turán graphs are named after Pál Turán, who used them to prove Turán's theorem, an important result in extremal graph theory. By the pigeonhole principle, every set of ''r'' + 1 vertices in the Turán graph includes two vertices in the same partition subset; therefore, the Turán graph does not contain a clique of size ''r'' + 1. According to Turán's theorem, the Turán graph has the maximum possible number of edges among all (''r'' + 1)-clique-free graphs with ''n'' vertices. Keevash and Sudakov (2003) show that the Turán graph is also the only (''r'' + 1)-clique-free graph of order ''n'' in which every subset of α''n'' vertices spans at least $\backslash frac(2\backslash alpha\; -1)n^2$ edges, if α is sufficiently close to 1. The Erdős–Stone theorem extends Turán's theorem by bounding the number of edges in a graph that does not have a fixed Turán graph as a subgraph. Via this theorem, similar bounds in extremal graph theory can be proven for any excluded subgraph, depending on the chromatic number of the subgraph. 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「Turán graph」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.