Phylogeny Numbers of Generalized Hamming Graphs

Abstract

For any positive integer n, we use [n] for the set $\{1,\ldots ,n\}$. For any integers $a_1,\ldots ,a_n \ge 2$ and $k\ge 1$, the generalized Hamming graph $\mathrm {H}_{a_1,\ldots ,a_n}^{k}$ is the graph with vertex set $[a_1]\times \cdots \times [a_n]$ in which two different vertices are adjacent if and only if their Hamming distance is at most k. We determine the phylogeny number of $\mathrm {H}_{a_1,\ldots ,a_n}^{1}$ and that of $\mathrm {H}_{m,m,m}^{2}$; we also calculate the phylogeny number of $\mathrm {H}_{a_1,\ldots ,a_n}^{n-1}$ when $a_1=\cdots =a_n$ 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.