1 7 Ju l 2 00 8 On perfect colorings of the halved 24 - cube ∗

A vertex coloring of a graph is said to be perfect with parameters (aij) k i,j=1 if for every i, j ∈ {1, ..., k} every vertex of color i is adjacent with exactly aij vertices of color j. We consider the perfect 2-colorings of the distance-2 graph of the 24-cube {0, 1}24 with parameters ((20+ c, 256− c)(c, 276− c)) (i.e., with eigenvalue 20). We prove that such colorings exist for all c from 1 to 128 except 1, 2, 4, 5, 7, 10, 13 and do not exist for c = 1, 2, 4, 5, 7.