In a fractional coloring, vertices of a graph are assigned measurable subsets of the real line and adjacent vertices receive disjoint subsets; the fractional chromatic number of a graph is at most if it has a fractional coloring in which each vertex receives a subset of of measure at least . We introduce and develop the theory of “fractional colorings with local demands” wherein each vertex “demands” a certain amount of color that is determined by local parameters such as its degree or the clique number of its neighborhood. This framework provides the natural setting in which to generalize degree-sequence type bounds on the independence number. Indeed, by Linear Programming Duality, all of the problems we study have an equivalent formulation as a problem concerning weighted independence numbers, and they often imply new bounds on the independence number.

中文翻译：

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局部要求的分数着色及其在独立数上的度数序列界限的应用


在分数着色中，图的顶点被分配实线的可测量子集，并且相邻顶点接收不相交的子集；如果图具有分数着色，其中每个顶点至少接收测度的子集，则该图的分数色数至多为 。我们引入并发展了“具有局部需求的分数着色”理论，其中每个顶点“需要”一定数量的颜色，该颜色由局部参数（例如其度数或其邻域的团数）决定。该框架提供了泛化独立数的度数序列类型界限的自然环境。事实上，通过线性规划对偶性，我们研究的所有问题都有一个与加权独立数问题等效的公式，并且它们通常意味着独立数的新界限。