Phylogeny Numbers of Generalized Hamming Graphs

For any positive integer n, we use [n] for the set {1,..., n}. For any integers a(1),..., a(n) >= 2 and k >= 1, the generalized Hamming graph H-a1,....,an(k) is the graph with vertex set [a1] x center dot center dot center dot x [an] in which two different vertices are adjacent if and only if their Hamming distance is at most k. We determine the phylogeny number of 1 a(1),...,a(n) and that of H-a1,....,an(n=1) we also calculate the phylogeny number ofHn-1 a(1),= ,an when a(1) = center dot center dot center dot = an is sufficiently large. In the course of establishing a lower bound estimate of phylogeny numbers, we make use of our former result on the rank of the rainbow inclusion matrices; our upper bound estimate comes from a concrete construction of a minimum size percolating set in a special bootstrap process.