Burning Hamming graphs

TL;DR

This paper provides an asymptotic estimate for the burning number of Hamming graphs $H(n,q)$ for fixed $q$, extending previous results from the binary case to larger alphabet sizes.

Contribution

It offers a concise proof that the burning number of $H(n,q)$ is approximately $(1-rac{1}{q})n$ with an error term, generalizing Alon's binary case result.

Findings

Burning number of $H(n,q)$ is $(1-rac{1}{q})n + O( ootrac{n ext{log} n}{})$ for fixed $q$.

Extension of previous binary case results to larger alphabet sizes.

Provides a short proof technique for asymptotic burning number estimates.

Abstract

The Hamming graph $H(n,q)$ is defined on the vertex set $[q]^n$ and two vertices are adjacent if and only if they differ in precisely one coordinate. Alon \cite{Alon} proved that the burning number of $H(n,2)$ is $\lceil\frac n2\rceil+1$. In this note we give a short proof of a fact that the burning number of $H(n,q)$ is $(1-\frac 1q)n+O(\sqrt{n\log n})$ for fixed $q\geq 2$ and $n\to\infty$.