Year: 2026
Type: preprint
Abstract: In Lazy Cops and Robbers, at most one cop moves on each cop turn. We study the lazy cop number of the generalized hypercube $Q(n,m)$, whose vertex set is ${\{0,1,\ldots,m\}}^n$. For each fixed integer $m\geq2$, we prove the asymptotic upper bound $$c_{\mathrm{L}}(Q(n,m))=O\!\left(\frac{{(m+1)}^n}{n^{3/2}}\right).$$ This result improves the upper bound of Sim, Tan, and Wong by a factor of $\log n$. The proof combines a moving dominating-set argument with an explicit dominating-set construction inside the support classes of each level. As a separate domination result, we show that, for fixed integers $m\geq2$ and $d\geq1$, the Hamming graph $K_m^{\square k}$ has a distance-$d$ dominating set of asymptotic size $O(m^k/k^d)$. This order is optimal up to a constant factor.
Source: arXiv (Cornell University)
Institution: National Institute of Technology Calicut
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7206196069