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Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory. It is a generalization of ordinary graph coloring. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different colors. In a fractional coloring however, a ''set'' of colors is assigned to each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common. Fractional graph coloring can be viewed as the linear programming relaxation of traditional graph coloring. Indeed, fractional coloring problems are much more amenable to a linear programming approach than traditional coloring problems. ==Definitions== A ''b''-fold coloring of a graph ''G'' is an assignment of sets of size ''b'' to vertices of a graph such that adjacent vertices receive disjoint sets. An ''a'':''b''-coloring is a ''b''-fold coloring out of ''a'' available colors. The ''b''-fold chromatic number χ''b''(''G'') is the least ''a'' such that an ''a'':''b''-coloring exists. The fractional chromatic number χf(''G'') is defined to be : Note that the limit exists because χ''b''(''G'') is ''subadditive'', meaning χ''a''+''b''(''G'') ≤ χ''a''(''G'') + χ''b''(''G''). The fractional chromatic number can equivalently be defined in probabilistic terms. χf(''G'') is the smallest ''k'' for which there exists a probability distribution over the independent sets of ''G'' such that for each vertex ''v'', given an independent set ''S'' drawn from the distribution, :. Some properties of χ''f''(''G''): : and :. Here n(''G'') is the order of ''G'', α(''G'') is the independence number, ω(''G'') is the clique number, and χ(''G'') is the chromatic number. 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「Fractional coloring」の詳細全文を読む スポンサード リンク
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