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

Haiying Zhou, Wai Chee SHIU*, Peter Che Bor Lam

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review


Suppose d is a positive integer. An L(d,1) -labeling of a simple graph G=(V,E) is a function f:V→N={0,1,2,⋯} such that |f(u)-f(v)|≥ d if dG(u,v)=1; and |f(u)-f(v)|≥ 1 if dG(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, λd(G), is the minimum of span over all L(d,1) -labelings of G. Whittlesey et al. proved that λ 2(Qn)≤ 2k+2k-q+1-2, where n≤ 2k-q and 1≤ q≤ k+1. As a consequence, λ2(Qn)≤ 2n for n≥ 3. In particular, λ 2(Q{2k-k-1)≤ 2k-1. In this paper, we provide an elementary proof of this bound. Also, we study the (1,1) -labeling number of Qn. A lower bound on λ1(Q n) are provided and λ1(Q2k-1) are determined.

Original languageEnglish
Pages (from-to)626-638
Number of pages13
JournalJournal of Combinatorial Optimization
Issue number3
Publication statusPublished - Oct 2014

Scopus Subject Areas

  • Computer Science Applications
  • Discrete Mathematics and Combinatorics
  • Control and Optimization
  • Computational Theory and Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Channel assignment problem
  • Distance two labeling
  • n -cube


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