Combinatorics Question

Given a complete graph with nodes N:

Whats the most uniform way to traverse it for any non-triangular number N?

This has driven me insane for a couple of years now, is there even a solution?

>>4
well then I haven't figured out how to formulate the question.
Oh yes I numbered the nodes... so I'll try again:

A numbered structure (represented by an array in memory) should be accessed in such a order that every element should follow every other element the same number of times and the distance distribution between elements is as uniform as possible. (Every possible distance has to be represented, upwards and downwards, so the "complete graph" would be traversed 2 times one time "forward" one "backwards") I somehow came up to the conclusion that if the number of elements is a triangular number complete uniform "traversal" is possible, else not.

Major problem is: I do not have enough math education to formulate *exactly* what I am looking for but know what it is.