I would like to know edges of what kind of graph may not be partitioned as triangles? As an example edges of one of these graphs $K_7 , k_{12} , K_{3,3,3} , K_{5,5,5}$ may not be partitioned as triangles but I don't know which one it is!
All came to my mind to solve this problem is, the number of edges must be $3x$ so I tried this for the above graphs: ($K_{n,n,n}$ is a 3-partitioned graph and each partition has $n$ vertices)
$$K_7: |E| = 21$$ $$K_{12}: |E| = 66$$ $$K_{3,3,3}: |E| = 27$$ $$K_{5,5,5}: |E| = 75$$
and sounds it doesn't work!
Hint: Show that if the edge set can be partitioned into triangles, then every vertex must have even degree.
(This rules out exactly one of your examples, although it might still be the case that the edge set of another one cannot be partitioned into triangles.)