Can I connect circuits for a dishwasher and sink disposal unit to the same breaker?
This is kind of a tricky question. For many years contractors connected the disposal and dishwasher on a split receptacle below the sink. Now with newer and larger appliances consuming more power there could be a problem. 

Also there has been some conversations in the AHJ and Code communities about NEC 400.12 (2) Uses not permitted - Where running through holes in walls ... Does it mean a fixed appliance electrical cords passing through a cabinet wall? Right now it's kind of up in the air and I'm betting there will be a code clarification coming up in newer revisions.

If you can perhaps the smarter move would be to extend the circuit from the disposal receptacle and install a separate receptacle behind the dishwasher. Or super safe install a separate circuit for each appliance. 

One last thing, both now need to be on GFCI protected circuits.

So make sure your GFCI protected, verify the circuit can hold the total load (that includes the start up surge of motors) using the nameplate data of the appliances and take it from there.

Hope this helps.