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


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 \(0 \le r \le 8\).


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



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.


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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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