Notes on $$L(1,1)$$ and $$L(2,1)$$ labelings for $$n$$ -cube

Suppose $$d$$ is a positive integer. An $$L(d,1)$$ -labeling of a simple graph $$G=(V,E)$$ is a function $$f:V\rightarrow \mathbb{N }=\{0,1,2,{\ldots }\}$$ such that $$|f(u)-f(v)|\ge d$$ if $$d_G(u,v)=1$$ ; and $$|f(u)-f(v)|\ge 1$$ if $$d_G(u,v)=2$$ . The span of an $$L(d,1)$$ -labeling $$f$$ is the absolute difference between the maximum and minimum labels. The $$L(d,1)$$ -labeling number, $$\lambda _d(G)$$ , is the minimum of span over all $$L(d,1)$$ -labelings of $$G$$ . Whittlesey et al. proved that $$\lambda _2(Q_n)\le 2^k+2^{k-q+1}-2,$$ where $$n\le 2^k-q$$ and $$1\le q\le k+1$$ . As a consequence, $$\lambda _2(Q_n)\le 2n$$ for $$n\ge 3$$ . In particular, $$\lambda _2(Q_{2^k-k-1})\le 2^k-1$$ . In this paper, we provide an elementary proof of this bound. Also, we study the $$L(1,1)$$ -labeling number of $$Q_n$$ . A lower bound on $$\lambda _1(Q_n)$$ are provided and $$\lambda _1(Q_{2^k-1})$$ are determined.