Let the hydrocarbon be $\mathrm{C_xH_y}$.

Use percentage of carbon Given that carbon is $90\%$ by mass and molar mass is $80\,\mathrm{g\,mol^{-1}}$.

Mass of carbon in 1 mole of hydrocarbon: $0.90 \times 80 = 72\text{ g}$

Since each mole of carbon contributes $12\text{ g}$, $x = \frac{72}{12} = 6$

So the formula contains $6$ carbon atoms: $\mathrm{C_6H_y}$.

Use molar mass to find hydrogen atoms Total molar mass is $80$: $12(6) + y = 80$ $72 + y = 80$ $y = 8$

Hence the molecular formula is: $\mathrm{C_6H_8}$

Calculate degree of unsaturation (DBE) For a hydrocarbon $\mathrm{C_nH_m}$, $\text{DBE} = \frac{2n+2-m}{2}$

Substituting $n=6$, $m=8$: $\text{DBE} = \frac{2(6)+2-8}{2} = \frac{12+2-8}{2} = \frac{6}{2} = 3$

Final answer The degree of unsaturation is: $\boxed{3}$