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.