In the Exact Cover by 3-Sets problem, we are given a set $X = \{x_1, x_2,\ldots, x_{3n}\}$ and a family of subsets $F = \{\{x_{i_1}, x_{i_2}, x_{i_3}\}\}$ of 3-element subsets of $X$. The question is then whether there exists a family of subsets $F'\subseteq F$ that forms an exact cover of $X$.
It is known that the problem remains NP-hard under some restrictions, e.g., to instances in which each element appears in exactly three subsets and $\lvert F_i\cap F_j\rvert\leq 1$ for all $F_i, F_j\in F$. (See this answer and its referenced paper.)
I would like to modify the problem in a slightly different way: the only restriction on the input is that we know that each element appears in exactly three subsets. The question, however, is this: can the family $F$ be partitioned into three disjoint families $F_1, F_2, F_3$ such that each $F_i$ is an exact cover?
Is this problem already known to be NP-hard/polynomial-time solvable? Another way to view it would be to consider a graph with vertices corresponding to subsets and edges being added if the respective subsets have a non-empty intersection. The problem then asks whether the graph is 3-colorable. Has 3-coloring (or k-coloring) perhaps been studied on this class of set-intersection graphs?
