Can I print each part's name on the optimisation diagram?

Yes you can. in MaxCut go to Layout Settings > Layout Grouping / Labelling.

Make sure "Panel Name" is selected for the screen diagrams and "Panel Name" selected for reports.

Then check the "Labelling" checkbox on how you would like the optimisation layouts grouped. 

Now when you run an optimisation you should see the name of each part on the optimisation layout and seperate diagrams when the parts are the same size but have different names.