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

## Abstract

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.

## Keywords

Channel assignment problem Distance two labeling \(n\)-cube## Notes

### Acknowledgments

This work is partially supported the FRG of Hong Kong Baptist University.

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