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.