These graphs
consist of
two wheels, Wm, with a common hub.
They could be composed as
We have shown that 2C3+K1 is not graceful, whereas 2Cn+K1 is for 3 < n < 11. We have found the number of non-isomorphic graceful labellings for n = 4 and 5; it takes too long to find all graceful labellings for larger values of n.
| Graph | A graceful labelling | Number of labellings |
| 2C3+K1 | - | 0 |
| 2C4+K1 | {0;1,3,16,12;5,14,8,15} | 44 |
| 2C5+K1 | {0;1,3,14,6,19;5,17,7,16,20} | 1216 |
| 2C6+K1 | {0;1,3,7,22,5,21;6,19,11,23,14,24} | - |
| 2C7+K1 | {0;1,3,8,20,27,6,24;9,22,26,10,25,11,28} | - |
| 2C8+K1 | {0;1,3,7,13,32,23,11,27;5,29,14,31,10,28,8,30} | - |
| 2C9+K1 | {0;1,3,7,12,26,6,35,13,31;8,33,10,34,15,32,11,27,36} | - |
| 2C10+K1 | {0;1,3,7,12,18,31,8,40,10,36;9,33,16,37,15,34,14,39,11,38} | - |
