Journal of Combinatorial Optimization

, Volume 28, Issue 3, pp 626–638 | Cite as

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

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


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.


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



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


  1. Bondy JA, Murty USR (1976) Graph theory with applications. MacMillan, New YorkzbMATHGoogle Scholar
  2. Griggs JR, Yeh RK (1992) Labeling graphs with a condition at distance two. SIAM J Discr Math 5:586–595CrossRefzbMATHMathSciNetGoogle Scholar
  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 68:1497–1514CrossRefGoogle Scholar
  7. Cozzens MB, Roberts FS (1982) \(T\)-colorings of graphs and the channel assignment problem. Congr Numer 35:191–208MathSciNetGoogle Scholar
  8. Cozzens MB, Wang DI (1984) The general channel assignment problem. Congr Numer 41:115–129MathSciNetGoogle Scholar
  9. Füredi Z, Griggs JR, Kleitman DJ (1969) Pair labelings with given distance. SIAM J Discr Math 2:491–499CrossRefGoogle Scholar
  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 7:605–613CrossRefzbMATHMathSciNetGoogle Scholar
  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 8:499–506CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

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

Personalised recommendations