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In graph theory, an area of mathematics, an equitable coloring is an assignment of colors to the vertices of an undirected graph, in such a way that *No two adjacent vertices have the same color, and *The numbers of vertices in any two color classes differ by at most one. That is, the partition of vertices among the different colors is as uniform as possible. For instance, giving each vertex a distinct color would be equitable, but would typically use many more colors than are necessary in an optimal equitable coloring. An equivalent way of defining an equitable coloring is that it is an embedding of the given graph as a subgraph of a Turán graph. There are two kinds of chromatic number associated with equitable coloring.〔.〕 The equitable chromatic number of a graph ''G'' is the smallest number ''k'' such that ''G'' has an equitable coloring with ''k'' colors. But ''G'' might not have equitable colorings for some larger numbers of colors; the equitable chromatic threshold of ''G'' is the smallest ''k'' such that ''G'' has equitable colorings for any number of colors greater than or equal to ''k''.〔Note that, when ''k'' is greater than the number of vertices in the graph, there nevertheless exists an equitable coloring with ''k'' colors in which all color classes have zero or one vertices in them, so every graph has an equitable chromatic threshold.〕 The Hajnal–Szemerédi theorem, posed as a conjecture by and proven by , states that any graph with maximum degree Δ has an equitable coloring with Δ + 1 colors. Several related conjectures remain open. Polynomial time algorithms are also known for finding a coloring matching this bound, and for finding optimal colorings of special classes of graphs, but the more general problem of deciding whether an arbitrary graph has an equitable coloring with a given number of colors is NP-complete. ==Examples== The star ''K''1,5 shown in the illustration is a complete bipartite graph, and therefore may be colored with two colors. However, the resulting coloring has one vertex in one color class and five in another, and is therefore not equitable. The smallest number of colors in an equitable coloring of this graph is four, as shown in the illustration: the central vertex must be the only vertex in its color class, so the other five vertices must be split among at least three color classes in order to ensure that the other color classes all have at most two vertices. More generally, observes that any star ''K''1,''n'' needs colors in any equitable coloring; thus, the chromatic number of a graph may differ from its equitable coloring number by a factor as large as ''n''/4. Because ''K''1,5 has maximum degree five, the number of colors guaranteed for it by the Hajnal–Szemerédi theorem is six, achieved by giving each vertex a distinct color. Another interesting phenomenon is exhibited by a different complete bipartite graph, ''K''2''n'' + 1,2''n'' + 1. This graph has an equitable 2-coloring, given by its bipartition. However, it does not have an equitable (2''n'' + 1)-coloring: any equitable partition of the vertices into that many color classes would have to have exactly two vertices per class, but the two sides of the bipartition cannot each be partitioned into pairs because they have an odd number of vertices. Therefore, the equitable chromatic threshold of this graph is 2''n'' + 2, significantly greater than its equitable chromatic number of two. 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「Equitable coloring」の詳細全文を読む スポンサード リンク
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