Hierarchical hyperbolicity of graphs of multicurves

We show that many graphs naturally associated to a connected, compact, orientable surface are hierarchically hyperbolic spaces in the sense of Behrstock, Hagen and Sisto. They also automatically have coarse median property defined by Bowditch. Consequences for such include distance formula analogous Masur Minsky's mapping class group, an upper bound on maximal dimension quasiflats, existence quadratic isoperimetric inequality. The structure gives rise simple criterion when Gromov hyperbolic.