Alon's transmitting problem and multicolor Beck--Spencer Lemma
Norihide Tokushige
TL;DR
This work determines the burning number of the Hamming graph $H(n,q)$ for $q\ge3$ by proving a lower bound via a multicolor discrepancy approach and providing a matching, explicit upper bound. The key method adapts the Beck--Spencer floating-variable technique to a multicolor setting using vector-coloring: vertices are encoded as $Q^n$ vectors so that $\mathbf{a}_i\cdot\mathbf{x}=(1-1/q)n-d(v_i,w)$, and a carefully constructed $\mathbf{x}$ yields a witness $w$ with $d(v_i,w)\ge m+1-i$ for all $i$, where $m=\left\lfloor(1-1/q)n\right\rfloor$. Consequently, for large $n$, the bounds are $\left\lfloor\left(1-\frac{1}{q}\right)n\right\rfloor+1 \le b(H(n,q)) \le \left\lfloor\left(1-\frac{1}{q}\right)n + \frac{q+1}{2}\right\rfloor$, and a simple construction provides the upper bound. The paper also discusses the case $q\ge n$ where the burning number becomes $n+1$, and outlines potential extensions of the distance-embedding lemma to antipodal/color-extensions, including concrete $q=3$ scenarios with implications for exact burning numbers.
Abstract
The Hamming graph $H(n,q)$ is defined on the vertex set $\{1,2,\ldots,q\}^n$ and two vertices are adjacent if and only if they differ in precisely one coordinate. Alon (1992) proved that for any sequence $v_1,\ldots,v_b$ of $b=\lceil\frac n2\rceil$ vertices of $H(n,2)$, there is a vertex whose distance from $v_i$ is at least $b-i+1$ for all $1\leq i\leq b$. In this note, we prove that for any $q\geq 3$ and any sequence $v_1,\ldots,v_b$ of $b=\lfloor(1-\frac1q)n\rfloor$ vertices of $H(n,q)$, there is a vertex whose distance from $v_i$ is at least $b-i+1$ for all $1\leq i\leq b$. Alon used a lemma due to Beck and Spencer (1983) which, in turn, was based on the floating variable method introduced by Beck and Fiala (1981) who studied combinatorial discrepancies. For our proof, we extend the Beck--Spencer Lemma by using a multicolor version of the floating variable method due to Doerr and Srivastav (2003).