# Partition into triangles in a 3-partite graphe

Let $G=(X\cup Y\cup Z,E)$ be a 3-partite graph such that:

1. $|X|=|Y|=|Z|=q$.
2. $2 \leq d(v) \leq 6$ for all $v \in X\cup Y\cup Z$, where $d(v)$ is the degree of v.
3. $\sum_{x \in X} d(x)=\sum_{y \in Y}d(y)=\sum_{z \in Z} d(z)$

Can we partition $G$ into $q$ vertex-disjoint triangles?

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.