Preheat oven to 350 degrees. Line muffin tins with cupcake liners (we made 18 cupcakes). Reserve.
In a large bowl, beat eggs for 2-3 minutes on medium speed until they turn a pale yellow. Add in sugar and beat until well combined. Add in mayonnaise, rum, and vanilla and beat until mixture is well blended.
In a medium bowl whisk together flour, baking soda, baking powder, salt and cocoa powder. Add flour mixture in 4 additions; mixing well between additions.
Divide batter evenly between prepared pans filling 3/4 full. Bake at 350 for 17 - 20 minutes or until a cake tester comes out clean. Allow to cool in pans on wire racks for 5 minutes. Then remove from pans, transfer to cooling racks and allow to cool completely before frosting.