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

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## 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.

## References

- Bondy JA, Murty USR (1976) Graph theory with applications. MacMillan, New YorkzbMATHGoogle Scholar
- Griggs JR, Yeh RK (1992) Labeling graphs with a condition at distance two. SIAM J Discr Math 5:586–595CrossRefzbMATHMathSciNetGoogle Scholar
- Chang GJ, Ke WT, Kuo D, Liu DF, Yeh RK (2000) On \(L(d,1)\)-labelings of graphs. Discr Math 220:57–66Google Scholar
- Roberts FS (1988) Private communication with J.R. GriggsGoogle Scholar
- Jonas K (1993) Graph coloring analogues wiwh a condition at distance two: \(L(2,1)\)-labeling and list \(\lambda \)-labeling. PhD Thesis, Dept. of Math., Univ. of South Carolina, ColumbiaGoogle Scholar
- Hale WK (1980) Frequency assignment: theory and applications. Proc IEEE 68:1497–1514CrossRefGoogle Scholar
- Cozzens MB, Roberts FS (1982) \(T\)-colorings of graphs and the channel assignment problem. Congr Numer 35:191–208MathSciNetGoogle Scholar
- Cozzens MB, Wang DI (1984) The general channel assignment problem. Congr Numer 41:115–129MathSciNetGoogle Scholar
- Füredi Z, Griggs JR, Kleitman DJ (1969) Pair labelings with given distance. SIAM J Discr Math 2:491–499CrossRefGoogle Scholar
- Raychaudhuri A (1985) Intersection assignment, \(T\)-coloring and powers of graphs. PhD Thesis, Dept. of Math., Rutgers Univ., New BrunswickGoogle Scholar
- Raychaudhuri A (1994) Further results on \(T\)-coloring and frequency assignment problems. SIAM J Discr Math 7:605–613CrossRefzbMATHMathSciNetGoogle Scholar
- Roberts FS (1986) \(T\)-colorings of graphs: recent resilts and open problems. Research Report RRR, RUTCOR, Rutgers Univ., New Brunswick, pp 7–86Google Scholar
- Roberts FS (1989) From garbage to rainbows: generalizations of graph colorings and their applications. In: Alavi Y, Chartrand G, Oellermann OR, Schwenk AJ (eds) Proceedings of the sixth international conference on the theory and applications of graphs. Wiley, New YorkGoogle Scholar
- Tesman B (1989) \(T\)-colorings, list \(T\)-colorings and set \(T\)-colorings of graphs. PhD thesis, Dept. of Math., Rutgers Univ., New BrunswickGoogle Scholar
- Whittlesey MA, Georges JP, Mauro DW (1995) On the \(\lambda \)-number of \(Q_n\) and related graphs. SIAM J Discr Math 8:499–506CrossRefzbMATHMathSciNetGoogle Scholar