A Coloring Problem in Hamming Spaces

TheR-domatic number of a graph is the maximum number of colors that can be used to color the vertices of the graph so that all vertices of the graph have at least one vertex of each color within distanceR.In this paper the problem of determining theR-domatic number of then-cube,P(n,R), is considered. The valueP(6,1) =5 is settled, and a conjecture by Laborde thatP(n, 1)≈n asntends to infinity is proved. Best known upper and lower bounds on theR-domatic number of then-cube are given forn≤16 andR≤7.