See the necessary groups (assuming that they have size at most )
Styling sufficient conditions:
(x, 2x) such that
(x, 2x, 3x) such that
(x, y) coprime such that or
(a, b, a+b) coprime such that (1) , (2) or (3)
(x, 2x, b) coprime such that
quadruples

For every list below, one of the numbers must be a speed (if 0 is never lonely). That means: if there is no vertex cover of size n, the lonely runner conjecture holds for n.

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Visualize pairs and then polish (or click two nodes to swap). Exclude these numbers (so triples can become pairs):

3-cycles:

5-cycles: