Here is what I have; please tell me if any steps have any error in reasoning.

The molar mass of CO_{2} is 44 g/mol and the molar mass of H_{2}O is 18 g/mol.

We can then say that the sum of the mass of the products can be represented with the expression of 18x + 44y = 372. Solving for integer solutions, the only integer solutions for the pair (x, y) is (6, 6). Therefore, the products side of the equation has the form 6CO_{2} + 6 H_{2}O, meaning that Compound A has 6 carbons and 12 hydrogens at the very least.

Two pieces of information verify my next step; in order for Compound A to be oxidized and lose 2 grams from its molar mass to form Compound B, two hydrogens must be lost. This can be done in the oxidation of an alcohol to either a ketone or an aldehyde. Therefore, there must be at least one oxygen.

One oxygen, with a molar mass of 16g, sums with 6 carbons (6 x 12 = 72) and 12 hydrogens (12 x 1 = 12) to 100g.

After determining that the formula of Compound A is C_{6}H_{12}O, the next step is to determine the structure. Assuming that there will be a hydroxide group to create the alcohol, effectively the formula is C_{6}H_{12} in terms of structure. This means that the structure must be an alkene or a cycloalkane. If it is an achiral compound and does not react to Br_{2}, then it can not be an alkene. Therefore, it must be a cycloalkane. If it is a cycloalkane, then there are a few options for its structure. However, only cyclohexanol fits the bill.

The structural formula for Compound A is (CH_{2})_{5}CHOH, and the structural formula for Compound B is (CH_{2})_{5}CO.