A correct solution was also received from Mike Fitzpatrick.

There is a unique smallest solution up to permutations of the cube. See the figure below. The vertex number, V, is 21 and the face number, F, is 28. The integers used are 1 to 6 and 8 to 13.

Here's one way to arrive at the solution. Let S denote the sum of all the edge numbers. Then 8V = 2S, since summing the vertex numbers for each of the eight corners sums each of the edges twice. Similarly, 6F = 2S. The equations 4V = S = 3F give that S must be divisible by 12. Since S is at least 1 + 2 + ... + 12 = 78, the smallest possible value for S is 84, which is the sum of the integers given above. The rest involves a small amount of trial and error.

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