Constrained shortest path problem
We can modify Dijkstra's algorithm to solve this problem too. For each vertex, in Dijkstra's algorithm, we store the shortest path to that vertex obtained so far. Here, we store $r+1$ different shortest paths per vertex (where the required colour sequence is $c_1,c_2,\ldots ,c_r$). In each vertex, the first path corresponds to shortest path so far, containing only whites. And for $i=1,2,\ldots ,r$, $(i+1)^{th}$ path is the shortest path so far, that contains vertices of colours $c_1,c_2,\ldots ,c_i$ (in that order) mixed with white vertices.

While modifying shortest paths for the new node, we will modify all $r+1$ paths if it is white and if it is of a different colour $c$, only those $(i+1)^{th}$ paths are modified which have $c_{i+1}=c$. And when the algorithm ends, the $(r+1)^{th}$ path in each vertex gives the required shortest path to that vertex.

Complexity will be $r$ times the complexity of Dijkstra's. Note that we have not assumed the colours in the sequence to be distinct (so we can get shortest paths with required sequence as red-black-red-blue too). Also, we can easily modify this to allow multiple vertices of same colour to occur together (i.e., red-red-black-blue can be considered same as red-black-blue).