Partition into triangles in a 3-partite graphe
A recent paper by Custic, Klinz, Woeginger "Geometric versions of the three-dimensional assignment problem under general norms", Discrete Optimization 18: 38-55 (2015) contains (and proves) the following proposition:


  Proposition 5.1: Partition into triangles on 6-regular tripartite graphs is NP-complete.


Now just follow the reduction in the proof with a NO-instance, and you will get a 6-regular tripartite graph on $3q$ vertices that cannot be partitioned into $q$ triangles.