Min path cover for a three-layer graph with all paths traversing all layers
I think this can be solved by reduction to a circulation problem.
Introduce a graph with source $a$ and sink $z$.  The edges are as follows:

All edges will have infinite capacity, lower bound 0, and cost 0 unless mentioned otherwise.
Add an edge $z \to a$ with cost 1.
Add edges $a \to c_i$ for each color $c_i$ and $s'_k \to z$ for each smell $s_k$.
For each allowable combination $c_i,t_j$, add an edge $c'_i \to t_j$, and each allowable combination $t_j,s_k$, add an edge $t'_j \to s_k$.
Add edges $c_i \to c'_i$, $t_j \to t'_j$, $s_k \to s'_k$ with lower bound 1.

Find the minimum cost solution to this circulation problem.  I think there exists an integral solution, and it can found in polynomial-time.
The solution corresponds to a collection of fruits, and the cost of the solution is the number of fruits required.  Each unit of flow corresponds to a fruit.  The structure of the graph ensures that each attribute is covered by at least one fruit.