Bírógyiköğlü–Hordijk–Leydold–Pisanski–Stadler conjecture on strong nodal domains of hypercubes

Let $H(n,2)$ be the Hamming graph with vertex set $\mathbb{Z}_2^n$, where two vertices are adjacent when they differ in exactly one coordinate. Its Laplacian has eigenvalue $2i$ for $0\leq i\leq n$. For an eigenfunction $f$, let $\mathrm{SND}(f)$ denote its number of strong nodal domains.

Bírógyiköğlü–Hordijk–Leydold–Pisanski–Stadler conjecture. For every $1\leq i\leq n-2$, there is an eigenfunction $f$ of $H(n,2)$ with eigenvalue $2i$ such that $\mathrm{SND}(f)=2$.

The conjecture extends the known result for $1\leq i\leq n/2$ and concerns the minimum possible number of strong nodal domains of hypercube eigenfunctions. The paper confirms it for $i$ up to approximately $2n/3$, while the remaining cases require a new approach.