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 number of vertex in color j adjacent to a fixed vertex in color i is aij , independent from 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-coloring of the 3-regular graphs of order at Most 8, that is, we determine a list of all color parameter matrices corresponding to perfect coloring of 3-regular graphs of order 4, 6 and 8.