# 3-coloring a graph with propositional formulas

I am trying to tackle a specific problem so any help would be greatly appreciated:

Let $$G = (V,E)$$ be an undirected graph with vertex set $$V$$ and edge set $$E$$. A 3-coloring of $$G$$ is a map $$\chi:V\to \{R,G,Y\}$$ such that if $$\{x,y\}\in E$$ then $$\chi(x)\neq \chi(y)$$. (Let $$R,G,Y$$ stand for red, blue, and yellow respectively).

Suppose $$n > 1$$ and let $$V_n = \{0,1,\cdots,n-1\}$$ and let $$G_n = (V_n,E_n)$$ be an undirected graph with vertex set $$V_n$$. For each $$i$$, $$0 \leq i < n$$ let $$R_i,B_i,Y_i$$ be propositional variables. We can think of $$i$$ being a node so $$R_i$$ says node $$i$$ has a color of red.

Give a propositional formula $$A_n$$ using the variables $$\{R_i,B_i,Y_i | 0 \leq i < n\}$$ such that $$A_n$$ is satisfiable iff $$G_n$$ has a 3-coloring. Do this in such a way that $$A_n$$ can be computed efficiently from $$G_n$$ (e.g. don't define $$A_n$$ to be $$R_1$$ if $$G_n$$ has a three coloring and ($$R_1 \wedge \neg R_1$$) otherwise).

My inclination for a question like this is to set up some sort of CNF formula, that is come up with a set of clauses that set out to take care of different properties. For instance I believe for something like this I need a clause that deals with the case that every node has a color, maybe one that deals with every node cannot have more than one color, and that every node cannot be the same color as an adjacent node? I am not really sure how to illustrate that last one or if there are cases that I am missing.

• What have you tried? Have you written down the requirements for a node labelling to be a 3-coloring? Have you tried translating those requirements into logic? Which ones did you run into problems with? – D.W. Apr 20 '14 at 22:07

The first can be formalized as $R_i \lor B_i \lor Y_i$ for all $i$.