Approximate graph colouring  with a promised upper bound on maximum independent set
Compute a maximum matching in the complement of the input graph. Every unmatched node must be in a different color class in any coloring. So: if you get at least cn matched edges, then the matching itself gives you a coloring with an upper bound of (1-c)n, and an approximation ratio of 64(1-c). If you don't get at least cn edges, then you get a lower bound of (1 - 2c)n colors and an approximation ratio of 1/(1-2c). Solving the equation 64(1-c) = 1/(1-2c) leads to an approximation ratio slightly larger than 32; see Sasho Nikolov's comment for the precise value.