I'm not sure if there is a neat relation between labelled (coloured) simple graphs and hypergraphs, however there is (from https://en.wikipedia.org/wiki/Hypergr

I'm not sure if there is a neat relation between labelled (coloured) simple graphs and hypergraphs, however there is (from https://en.wikipedia.org/wiki/Hypergraph https://en.wikipedia.org/wiki/Hypergraph) : > Hypergraphs can be viewed as incidence structures. In particular, there is a bipartite "incidence graph" or "Levi graph" corresponding to every hypergraph Where the relation is one-to-one, so there is a Levi graph for every hypergraph and the reverse is also true.