We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of rank functions of matroids, and for which the minimum number of matroids required in such sums is the chromatic number of the line graph of the hypergraph. This result motivates introducing chromatic numbers and chromatic polynomials for polymatroids. We show that the chromatic polynomial of any 2-polymatroid is a rational multiple of the chromatic polynomial of some graph. We also find the excluded minors for the minor-closed class of polymatroids that can be written as sums of rank functions of matroids that form a chain of quotients.

我们介绍一些想法，这些想法是对多类拟曲线和图形着色之间许多已知联系的补充。给定一个满足某些条件的超图，我们构造以阶函数给出的多拟阵，可以将其写为拟阵阵的阶次函数之和，为此，求和的最小拟阵数为线形图的色数。超图。该结果促使引入用于多拟阵的色数和色多项式。我们表明，任何2多重拟阵的色多项式是某个图的色多项式的有理倍数。我们还找到了多封闭类的未封闭类的被排除的未成年人，该类可写为形成商链的拟阵的秩函数之和。