Perfect 4-Colorings of the 3-Regular Graphs of Order 10

The perfect m-coloring with matrix A = [aij ]i;j2f1;


;mg of a graph G = (V;E) withf1;


;mg color is a vertices coloring of G with m-color so that number of vertex in color j adjacent to a xed vertex in color i is aij , independent of the choice of vertex in color i. The matrix A = [aij ]i;j2f1;


;mg is called the parameter matrix.We study the perfect 4-colorings of the 3-regular graphs of order 10, that is, we determine a list of all color parameter matrices corresponding to perfect colorings of 3-regular graphs of order 10.