#### Document Type

Journal Article

#### Department/Unit

Department of Mathematics

#### Title

A new approach to the L(2,1) -labeling of some products of graphs

#### Language

English

#### Abstract

The frequency assignment problem is to assign a frequency which is a nonnegative integer to each radio transmitter so that interfering transmitters are assigned frequencies whose separation is not in a set of disallowed separations. This frequency assignment problem can be modelled with vertex labelings of graphs. An L(2, 1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that vert f(x) - f(y)\vert ≥ 2 if d(x,y) = 1 and vert f(x) - f(y)≥ 1 if d(x,y) = 2, where d(x,y) denotes the distance between x and y in G. The L(2, 1)-labeling number λ (G)of G is the smallest number k such that G has an L(2, 1)-labeling with max f(v):v\in V(G) = k. In this paper, we develop a dramatically new approach on the analysis of the adjacency matrices of the graphs to estimate the upper bounds of λ-numbers of the four standard graph products. By the new approach, we can achieve more accurate results and with significant improvement of the previous bounds. © 2008 IEEE.

#### Keywords

Cartesian product, Channel assignment, Direct product, L(2, 1)-labeling, Lexicographic product, Strong product

#### Publication Date

2008

#### Source Publication Title

IEEE Transactions on Circuits and Systems II: Express Briefs

#### Volume

55

#### Issue

8

#### Start Page

802

#### End Page

805

#### Publisher

Institute of Electrical and Electronics Engineers

#### DOI

10.1109/TCSII.2008.922450

#### Link to Publisher's Edition

#### ISSN (print)

15497747

#### Recommended Citation

Shiu, Wai Chee,
Zhendong Shao,
Kin Keung Poon,
and
David Zhang.
"A new approach to the L(2,1) -labeling of some products of graphs."
*IEEE Transactions on Circuits and Systems II: Express Briefs*
55.8
(2008): 802-805.