Final answer:
To guarantee all brown-coated offspring from mating with a yellow lab with genotype bbee, the other dog must have the genotype bbEE, as it has the necessary alleles for brown coat color without the epistatic yellow genotype.
Step-by-step explanation:
To produce offspring with brown coats while mating with a yellow lab (genotype bbee), we must find a genotype that does not carry the recessive ee combination that causes the yellow coat color, and which also carries the homozygous recessive bb for brown coloration. The correct genotype is bb. None of the other options provided (BBEE, BbEe, bbee) will result in all brown-coated offspring when mated with a yellow lab (bbee) because they either have the dominant B allele which would result in some black-coated offspring (BBEE, BbEe), or they have the ee genotype (bbee) which is epistatic and would result in yellow-coated offspring.
If a dominant E allele is present, a B allele results in a black coat, whereas the homozygous bb genotype results in a brown coat. When mated with a yellow lab with genotype bbee, the genotype of dog that would produce offspring with brown coats (all of them) is bbEE, where both copies of the B gene are the recessive bb genotype and at least one copy of the E gene is a dominant E allele.