Journal of Combinatorial Optimization

, Volume 28, Issue 3, pp 626–638

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

• Haiying Zhou
• Wai Chee Shiu
• Peter Che Bor Lam
Article

## 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

1. Bondy JA, Murty USR (1976) Graph theory with applications. MacMillan, New York
2. Griggs JR, Yeh RK (1992) Labeling graphs with a condition at distance two. SIAM J Discr Math 5:586–595
3. Chang GJ, Ke WT, Kuo D, Liu DF, Yeh RK (2000) On $$L(d,1)$$-labelings of graphs. Discr Math 220:57–66Google Scholar
4. Roberts FS (1988) Private communication with J.R. GriggsGoogle Scholar
5. 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
6. Hale WK (1980) Frequency assignment: theory and applications. Proc IEEE 68:1497–1514
7. Cozzens MB, Roberts FS (1982) $$T$$-colorings of graphs and the channel assignment problem. Congr Numer 35:191–208
8. Cozzens MB, Wang DI (1984) The general channel assignment problem. Congr Numer 41:115–129
9. Füredi Z, Griggs JR, Kleitman DJ (1969) Pair labelings with given distance. SIAM J Discr Math 2:491–499
10. Raychaudhuri A (1985) Intersection assignment, $$T$$-coloring and powers of graphs. PhD Thesis, Dept. of Math., Rutgers Univ., New BrunswickGoogle Scholar
11. Raychaudhuri A (1994) Further results on $$T$$-coloring and frequency assignment problems. SIAM J Discr Math 7:605–613
12. Roberts FS (1986) $$T$$-colorings of graphs: recent resilts and open problems. Research Report RRR, RUTCOR, Rutgers Univ., New Brunswick, pp 7–86Google Scholar
13. 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
14. Tesman B (1989) $$T$$-colorings, list $$T$$-colorings and set $$T$$-colorings of graphs. PhD thesis, Dept. of Math., Rutgers Univ., New BrunswickGoogle Scholar
15. Whittlesey MA, Georges JP, Mauro DW (1995) On the $$\lambda$$-number of $$Q_n$$ and related graphs. SIAM J Discr Math 8:499–506

## Authors and Affiliations

• Haiying Zhou
• 1
• 2
• Wai Chee Shiu
• 1
• Peter Che Bor Lam
• 2
1. 1.Department of MathematicsHong Kong Baptist UniversityHong KongChina
2. 2.Division of Science and TechnologyBNU-HKBU United International CollegeZhuhaiChina