## Abstract

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 d_{G}(u,v)=1; and |f(u)-f(v)|≥ 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, λ_{d}(G), is the minimum of span over all L(d,1) -labelings of G. Whittlesey et al. proved that λ _{2}(Q_{n})≤ 2^{k}+2^{k-q+1}-2, where n≤ 2^{k}-q and 1≤ q≤ k+1. As a consequence, λ_{2}(Q_{n})≤ 2n for n≥ 3. In particular, λ _{2}(Q_{{2k-k-1})≤ 2^{k}-1. In this paper, we provide an elementary proof of this bound. Also, we study the (1,1) -labeling number of Q_{n}. A lower bound on λ_{1}(Q _{n}) are provided and λ_{1}(Q_{2k-1}) are determined.

Original language | English |
---|---|

Pages (from-to) | 626-638 |

Number of pages | 13 |

Journal | Journal of Combinatorial Optimization |

Volume | 28 |

Issue number | 3 |

DOIs | |

Publication status | Published - 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