Perfect 3-colorings of Heawood graph

Perfect coloring is a generalization of the notion of completely regular codes, given by Delsarte. A perfect m-coloring of a graph G with m colors is a partition of the vertex set of G into m parts A1, . . . ,Am such that, for all i, j ∈ {1, . . . ,m}, every vertex of Ai is adjacent to the same number of vertices, namely, aij vertices, of Aj . The matrix A = (aij)i, j ∈ {1, 2, ,m}, is called the parameter matrix. We study the perfect 3-colorings (also known as the equitable partitions into three parts) of the Heawood graph. In particular, we classify all the realizable parameter matrices of perfect 3-colorings for the Haywood graphs.