I have found a representation of the graph coloring problem as an ILP. Given a graph $G = (V, E)$.
Let $C$ represent the set of colors. Let $w_c$ be a binary variable that is $1$ if the color $c$ is used and $0$ otherwise. Let $x_{v,c}$ be a binary variable that is $1$ if the vertex $v$ is colored by $c$.
ILP $$ \min \sum_{c \in C} w_c \\ \text{s.t} \sum_{c \in C} x_{v, c} = 1 \; \forall \; v \in V \\ x_{u, c} + x_{v, c} \leq w_c \; \forall \; (u, v) \in E, \; \forall \; c \in C \\ x_{v,c}, w_c \in \{0, 1\} \; \forall v \in V, \; \forall \; c \in C $$
I want to find the dual of the LP relaxation.
The problem I think, is that there are multiple binary values instead of one which I am used to for linear programming. I would assume that for the LP relaxation, the constraint for both $x_{v, c}$ and $w_c$ is set to greater than or equal to $0$, that is the LP relaxation will be:
$$ \min \sum_{c \in C} w_c \\ \text{s.t} \sum_{c \in C} x_{v, c} = 1 \; \forall \; v \in V \\ x_{u, c} + x_{v, c} \leq w_c \; \forall \; (u, v) \in E, \; \forall \; c \in C \\ 1 \geq x_{v,c}, w_c \geq 0 \; \forall v \in V, \; \forall \; c \in C $$
From here I am not really sure how to proceed. Any help would be appreciated:)