For every 2-polymatroid, the chromatic polynomial counting its matroid decompositions equals a rational multiple of the chromatic polynomial of some graph, and special hypergraphs realize the graph coloring number as the minimal decomposition size.
Lemos, Uniqueness of the decomposition of the rank fun ction of a 2-polymatroid, Discrete Math
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Decomposable polymatroids and connections with graph coloring
For every 2-polymatroid, the chromatic polynomial counting its matroid decompositions equals a rational multiple of the chromatic polynomial of some graph, and special hypergraphs realize the graph coloring number as the minimal decomposition size.