In many practical applications the underlying graph must be as equitable colored as possible. A coloring is called equitable if the number of vertices colored with each color differs by at most one, and the least number of colors for which a graph has such a coloring is called its equitable chromatic number. We introduce a new integer linear programming approach for studying the equitable coloring number of a graph and show how to use it for improving lower bounds for this number. The two stage method is based on finding or upper bounding the maximum cardinality of an equitable color class in a valid equitable coloring and, then, sequentially improving the lower bound for the equitable coloring number. The computational experiments were carried out on DIMACS graphs and other graphs from the literature.

中文翻译：

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改善公平色数的下限

在许多实际应用中，底层图形必须尽可能均匀地着色。如果用每种颜色着色的顶点数最多相差一个，则着色称为公平的，并且图具有这种着色的最少颜色数称为其公平色数。我们引入了一种新的整数线性规划方法来研究图形的公平着色数，并展示如何使用它来改善该数字的下界。两阶段方法基于在有效的公平着色中找到或确定公平颜色类别的最大基数上限，然后依次改进公平着色数的下限。计算实验是在 DIMACS 图和文献中的其他图上进行的。