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

The perfect 𝑚-coloring with matrix 𝐴 = [𝑎𝑖𝑗] 𝑖,𝑗 ∈ {1,...,m} of a graph 𝐺 = (𝑉, 𝐸) with {1, . . . , 𝑚} color is a vertices 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 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 10, that is, we determine a list of all color parameter matrices corresponding to perfect colorings of 3-regular graphs of order 10.