We have all the possible 3-grams of lower case letters (A = {a, b, c, ..., z}):
A³ = A × A × A = 
{
  (a, a, a), 
  (a, a, b), 
  (a, a, c), 
    ..., 
  (a, a, z), 
  (b, a, a), 
    ..., 
  (z, z, y), 
  (z, z, z), 
}

|A³| = 26³ = 17576

• If we try to merge aaa with bbb we have to append bbb to the end of aaa, because they don't overlap:
  merge(aaa, bbb) = aaabbb

• If we merge aaa with aab we can create a smaller string:
  merge(aaa, aab) = aaab

In the best case, if we manage to always do a second kind of merge, we will be extending the resulting string by 1 character. There are 17576 triplets, so there can be at most 17575 single character extensions. This will result in 17575 + 3 = 17578 characters string.

In the worst case, we will append all of the 3-grams to each other, thus creating a 17576 * 3 = 52728 characters string.

How to combine all of them to get the total string of the minimum length?

Picture to attract people :)