Burning Hamming graphs

The Hamming graph H(n, q) is defined on the vertex set \([q]^n\) ${[q]}^n$ and two vertices are adjacent if and only if they differ in precisely one coordinate. Alon (Disc Appl Math 37(38):9–11, 1992) proved that the burning number of H(n, 2) is \(\lceil \frac{n}{2}\rceil +1\) $\lceil \frac{n}{2}\rceil +1$ . In this note we give a short proof of a fact that the burning number of H(n, q) is \((1-\frac{1}{q})n+O(\sqrt{n\log n})\) $(1-\frac{1}{q})n+O(\sqrt{nlogn})$ for fixed \(q\ge 2\) $q\ge 2$ and \(n\rightarrow \infty \) $n\to \infty$ .