The Hamming graph efficient domination divisibility conjecture

Let $q=p^b$ for a prime $p$, and let $H(q,d)$ denote the Hamming graph with alphabet size $q$ and dimension $d$. An efficient $(1,k)$-dominating function is a function of the type specified in the paper whose domination condition is attained efficiently. Suppose that

$$(d-1)q+1=p^a m,$$

where $m$ is not divisible by $p$. Hamming graph efficient domination conjecture. The graph $H(q,d)$ has an efficient $(1,k)$-dominating function if and only if $1\leq kleq (q-1)d+1$ and $k$ is a multiple of $m$. This conjecture asserts that the necessary divisibility condition is sufficient for Hamming graphs when $q$ is a prime power; its resolution is not supplied in the source.