Minimizing maximal adjacent integer sum on a circle

Question

Arrange $\{1,2,\cdots,n\}$ on a circle. What are the arrangements that minimize the maximal sum of all adjacent $k$ integers? Some nontrivial examples of $(n,k)$ are welcome. A random algorithm that gives a probabilistic bound would be great.

Answer 1

we don't want the two largest numbers in the set: which is n and n-1 to be adjacent to each other: You can quickly figure out the pattern that the lower bound of the maximum adjacent sum is n+2:

The general pattern is to start with n - the greatest number - on the very top position and make sure that n-1 isn't beside n, and then if you can n-2 isn't beside either n or n-1 and n-3 isn't beside any of those and so on. I placed them exactly 2 spots away in the clockwise direction (although you could use the Counter-Clockwise convention instead).