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Journal of Combinatorial Optimization

, Volume 31, Issue 2, pp 447–462 | Cite as

\(L(2,1)\)-labeling for brick product graphs

  • Zehui Shao
  • Jin Xu
  • Roger K. Yeh
Article

Abstract

Let \(G=(V, E)\) be a graph. Denote \(d_G(u, v)\) the distance between two vertices \(u\) and \(v\) in \(G\). An \(L(2, 1)\)-labeling of \(G\) is a function \(f: V \rightarrow \{0,1,\cdots \}\) such that for any two vertices \(u\) and \(v\), \(|f(u)-f(v)| \ge 2\) if \(d_G(u, v) = 1\) and \(|f(u)-f(v)| \ge 1\) if \(d_G(u, v) = 2\). The span of \(f\) is the difference between the largest and the smallest number in \(f(V)\). The \(\lambda \)-number of \(G\), denoted \(\lambda (G)\), is the minimum span over all \(L(2,1 )\)-labelings of \(G\). In this article, we confirm Conjecture 6.1 stated in X. Li et al. (J Comb Optim 25:716–736, 2013) in the case when (i) \(\ell \) is even, or (ii) \(\ell \ge 5\) is odd and \(0 \le r \le 8\).

Keywords

L(2, 1)-labeling Brick product graph Graph labeling Frequency assignment problem 

Notes

Acknowledgments

This work was supported by the National Natural Science Foundation of China under the grant 61309015 and National Science Council grant, NSC-98-2115-M-035-002-MY3.

References

  1. Bodlaender HL, Kloks T, Tan RB, van Leeuwen J (2004) The \(L\)(2, 1)-labeling problem on graphs. Comput J 47:193–204Google Scholar
  2. Calamoneri T (2011) The \(L(h, k)\)-labeling problem: a updated aurvey and annotated bibliography. Comput J 54:1344–1371Google Scholar
  3. Fiala J, Golovach PA, Kratochvíl JK (2005) Distance constrained labelings of graphs of bounded treewidth. In Proceedings of the 32th ICALP 9:360–372Google Scholar
  4. Griggs JR, Yeh RK (1992) Labelling graphs with a condition at distance 2. SIAM J Discret Math 308:586–595MathSciNetCrossRefGoogle Scholar
  5. Hale WK (1980) Frequency assignment: theory and applications. Proc IEEE 68:1497–1514CrossRefGoogle Scholar
  6. Jha PK (2001) Optimal \(L\)(2, 1)-labeling of strong products of cycles. IEEE Trans Circ Sys I Fund Theory Appl 48:498–500Google Scholar
  7. Klavžar S, Vesel A (2003) Computing graph invariants on rotagraphs using dynamic algorithm approach: the case of (2,1)-colorings and independence numbers. Discret Appl Math 129:449–460CrossRefMATHGoogle Scholar
  8. Korz̆e D, Vesel A (2005) \(L\)(2, 1)-labeling of strong products of cycles. Inf Process Lett 94:183–190Google Scholar
  9. Kratochvíl J, Kratsch D, Liedloff M (2007) Exact algorithms for \(L\)(2, 1)-labeling of graphs. MFCS 2007. LNCS 4708:513–524Google Scholar
  10. Li X, Mak-Hau V, Zhou S (2013) The \(L(2, 1)\)-labelling problem for cubic cayley graphs on dihedral groups. J Comb Optim 25:716–736Google Scholar
  11. Schwarz C, Troxell DS (2006) \(L(2, 1)\)-labelings of products of two cycles. Disc Appl Math 154:1522–1540Google Scholar
  12. Whittlesey MA, Georges JP, Mauro DW (1995) On the \(\lambda \) number of \(q_n\) and related graphs. SIAM J Discret Math 8:499–506MathSciNetCrossRefMATHGoogle Scholar
  13. Yeh RK (2006) A survey on labeling graphs with a condition at distance two. Discret Math 306:1217–1231CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Information Science and TechnologyChengdu UniversityChengduChina
  2. 2.Key Laboratory of Pattern Recognition and Intelligent Information ProcessingInstitutions of Higher Education of Sichuan ProvinceChengduChina
  3. 3.School of Electronic Engineering and Computer SciencePeking UniversityBeijingChina
  4. 4.Department of Applied MathematicsFeng Chia UniversityTaichungTaiwan

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