An unknown hydrocarbon $A$ with the formula $\mathrm{C}_{6} \mathrm{H}_{12}$ reacts with 1 molar equivalent of $\mathrm{H}_{2}$ over a palladium catalyst. Hydrocarbon $\mathrm{A}$ also reacts with $\mathrm{OsO}_{4}$ to give diol $\mathrm{B}$. When oxidized with $\mathrm{KMnO}_{4}$ in acidic solution, A gives two fragments. One fragment is propanoic acid, $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CO}_{2} \mathrm{H},$ and the other fragment is ketone $\mathrm{C}$. What are the structures of $\mathrm{A}, \mathrm{B},$ and C? Write all reactions, and show your reasoning.