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:Vrightarrow 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 (nle 2^k-q) and (1le qle k+1) . As a consequence, (lambda _2(Q_n)le 2n) for (nge 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.