So, what you are looking for is the formulas and word equations of combustion:
Complete Combustion occurs when a fuel burns in plenty of oxygen.
Word Equation is methane + oxygen --> carbon dioxide + oxygen
Incomplete combustion's word equation =
fuel + oxygen --> carbon monoxide + water
fuel + oxygen --> carbon + water
Toxic gas (carbon monoxide) and soot (carbon) is made during incomplete combustion.
Balanced symbol equations:
Complete combustion:
CH4 + 2O2 --> CO2 + 2H2O
Incomplete combustion:
2CH4 + 3O2 --> 2CO + 4H20 when carbon monoxide is released
OR
CH4 + O2 --> C + 2H2O when carbon is released
As you can see complete combustion occurs when a fuel burns in plenty of oxygen, which is why it releases more energy than incomplete combustion.
Therefore, incomplete combustion doesn't release oxygen only but instead water.
This means that incomplete combustion occurs due to a lack of oxygen in the air and that would release more carbon monoxide than carbon dioxide.