What is the relation between diabetes and hemoglobin?

No direct one: Diabetes in and of itself doesn't cause any change in hemoglobin. There is, however, glycohemoglobin, or hemoglobin A1c that is a direct measure of one's glucose control. That has directly to do with diabetes. Plain hemoglobin is usually affected in patients with kidney failure as their erythropoetin production fails. Also, anemia of chronic disease given many comorbidities with dm is another one.

Nothing sort of: There is a form of hemoglobin which is called glycated hemoglobin. The amount or % of this hemoglobin can and is used as a measurement of average blood sugar over time (life of the red cell). This is the a1c test. In non-dabetics the value is 5.5% or below. I like to see diabetics in the 6.5% range to consider them good control. This is a simple answer with a lot more to it. Google it.