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 GG and positive integers B1,…,BkB1,…,Bk, and have to determine whether there exists an anticoloring of GG such that BjBj vertices are colored in color jj, 1≤j≤k1≤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.