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(Under construction)
These are like double wheel graphs, but the vertices of the two wheels are joined pairwise. They could alternatively be thought of as a prism CnxP2 with every vertex joined to a common point.
Since every vertex has even degree, these graphs are not graceful if the number of edges is congruent to 1 or 2 mod 4, i.e. if n is congruent to 1 or 2 mod 4.
They are graceful for n = 3, 4, 7, 8; we have not yet been able to test any larger values of n. We conjecture that they are graceful whenever n is congruent to 0 or 3 mod 4.
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| Graph | A graceful labelling | Number of labellings |
| (C3xP2)+K1 | {0;6,15,11;13,1,3} | 10 |
| (C4xP2)+K1 | {0;1,7,9,19;16,20,17,5} | 90 |
| (C7xP2)+K1 | {0;1,3,7,12,33,13,31;25,35,26,34,6,29,14} | - |
| (C8xP2)+K1 | {0;1,3,7,12,36,6,38,18;11,37,16,35,8,39,25,40} | - |
These graphs consist of three wheels Wn with a common hub. 3C3+K1 is not graceful; but the graph is graceful, with a large number of graceful labellings, for n = 4, 5, 6.
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| Graph | A graceful labelling | Number of labellings |
| 3C3+K1 | - | 0 |
| 3C4+K1 | {0;1,3,7,22;5,18,8,24;6,20,11,23} | > 2000 |
| 3C5+K1 | {0;1,3,11,26,14;4,20,29,10,27;6,28,7,25,30} | > 100 |
| 3C6+K1 | {0;1,3,7,12,20,30;6,32,13,36,9,34;11,33,15,31,14,35} | > 100 |