Lattice spanners of low degree

Let ${\delta }_0(P,k)$ denote the degree $k$ dilation of a point set $P$ in the domain of plane geometric spanners. If $\mathrm{\Lambda }$ is the infinite square lattice, it is shown that $1+\sqrt{2} \leq \delta_0(\Lambda,3) \leq (3+2\sqrt2) , 5^{-1/2} = 2.6065\ldots$and$\delta_0(\Lambda,4) = \sqrt{2}$\mathrm{.}If$\Lambda$istheinfinitehexagonallattice,itisshownthat$\delta_0(\Lambda,3) = 1+\sqrt{3}$and$\delta_0(\Lambda,4) = 2$. All our constructions are planar lattice tilings constrained to degree $3$ or $4$.