simplices surfaces edges and loops

Question

I am trying to orient the edges of higher dimensional simplices so that there are a maximal amount of loops consisting of only 3 edges.I am in particular working with simplices of even dimensions, eg. 4, 6, 8, so that each vertex can have the same number of incoming and outgoing edges (Practically I am using this to model audio reverberation networks) I tried to draw some examples in 4 and 6 dimensions, respectively 10 and 35 triangles. For 4 dimensions I get in both cases 5 loops of 3 edges and 5 triangles not looping, For 6 dimensions I get in both cases 14 loops and 21 triangles which do not have a loop. Looks a bit like some group theory is lurking beneath this. Any hints or keywords of how to optimize/ systematize this problem is most welcome. Thank you in advance Vilbjørg