The halved cube graph for n is isomorphic to the Hamming radius-2 graph for n - 1. We work here in the Hamming graph. The integers below indicate the dominating sets via inverse images. For example, if n = 4, we work with strings of length 3 in the Hamming graph. The domatic set is derived from {1,2,3,4,4,3,2,1} is {1, 8}, {2, 7}, {3, 6}, {4, 5} Subtracting 1 because we wish the vertices to be from 0 to 7 gives {0, 7}, {1, 6}, {2, 5}, {3, 4} and writing these in binary gives the four dominating sets in terms of bit-strings {000, 111} {001, 101} {010, 101} {011, 100} The following list gives the optimal domatic sets up to n = 8 (so up to HammingGraph(7,2). 3: {1, 2, 3, 4} 4: {1,2,3,4,4,3,2,1} 5: {1,2,3,4,5,6,7,8,8,7,6,5,4,3,2,1} 6: {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1} 7: {1,14,10,2,16,7,11,13,8,6,12,12,6,10,5,8,3,7,15,4,15,2,3,5,13,9,11,16,4,9,14,1,5,11, 9,1,14,13,9,4,4,15,7,3,16,3,2,15,10,8,6,16,12,12,8,6,2,5,13,14,1,11,7,10} 8: {1,2,3,4,5,6,7,4,8,9,10,4,4,11,9,12,13,14,1,14,5,13,15,11,10,12,12,16,17,15,9,3, 8,18,14,3,12,11,16,6,10,17,3,5,16,17,18,1,18,9,10,18,13,7,15,7,6,1,2,13,2,11,15, 8,16,8,17,6,5,10,18,2,15,7,18,15,14,3,13,2,11,9,11,17,1,16,8,8,7,3,6,5,12,18,6,14, 12,15,7,13,12,6,9,17,8,13,11,1,7,14,16,1,4,16,17,12,3,5,2,10,2,9,5,7,14,4,4,10}