Anticoloring of the rook's graph

An anticoloring of a graph is a partial coloring of the vertices in which no two adjacent vertices are colored in distinct colors. In the basic anticoloring problem, we are given a graph G and positive integers B₁,⋯, B_k, and have to determine whether there exists an anticoloring of G such that B_j vertices are colored in color j, 1 ≤ j ≤ k. This problem is known to be NP-complete, even for two colors. We deal with the anticoloring problem on the rook's graph. In general, we are able to provide sub-linear algorithms. In some particular cases, we give an explicit formula for the optimal solution.