I'm struggling to answer a past paper question, which asks to prove that the defined problem is in the polynomial complexity class(P). The question is mentioned below
The only strategy I can come up with for this question is to somehow reduce the problem to the 2-color decision problem(determine whether or not the vertices of a graph can be colour with two colours with no vertex adjacent to another vertex with the same colour), and I already know that the 2-color decision problem is in P.
The problem with this strategy is that I will need to drop one of my three colours in order to reduce the problem to a 2-color decision problem, also I can't feed the same graph into the 2-color algorithm(possibly by merging two colours into one, e.g replace blue and yellow, with green), since a graph containing triangles can be coloured using the restricted 3 colours, however any graph containing a triangle cannot be 2-colourable. So if my strategy is correct then how can I reconstruct my original graph so that it can be fed into the algorithm for the 2-color decision problem.
Hint: You can reduce the problem to 2-CNF-SAT.
Have clauses such as $B_{17}\lor R_{17}$, $\neg B_{17} \lor \neg R_{17}$, and $\neg B_{17}\lor \neg B_{42}$.