perfect 4-colorings of the 3-regular graphs of order at most 8

The perfect m-coloring with matrix a = [aij ]i,j∈{1,··· ,m} of a graph g = (v, e) with {1, · · · , m} color is a vertex coloring of g with m-color so that the number of vertices in color j adjacent to a fixed vertex in color i is aij , independent of the choice of vertex in color i. the matrix a = [aij ]i,j∈{1,··· ,m} is called the parameter matrix. we study the perfect 4-colorings of the 3-regular graphs of order at most 8, that is, we determine a list of all color parameter matrices corresponding to perfect 4-colorings of 3-regular graphs of orders 4, 6, and 8.