A New Approach to the L(2,1) -Labeling of Some Products of Graphs

• Published in: IEEE transactions on circuits and systems. II, Express briefs Vol. 55; no. 8; pp. 802 - 805
• Main Authors: Wai Chee Shiu, Zhendong Shao, Kin Keung Poon, Zhang, D.
• Format: Journal Article
• Language: English
• Published: IEEE 01.08.2008
• Subjects: 1) -labeling; Cartesian product; Channel assignment; Contracts; direct product; Frequency; Graph theory; L; Labeling; lexicographic product; Mathematics; Radio transmitters; Standards development; strong product; Upper bound; User-generated content
• ISSN: 1549-7747; 1558-3791
• DOI: 10.1109/TCSII.2008.922450

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 |f(x)-f(y)|ges2 if d(x, y)=1 and |f(x)-f(y)|ges1 if d(x, y)=2, where d(x, y) denotes the distance between x and y in G. The L(2, 1)-labeling number lambda(G) of G is the smallest number k such that G has an L(2, 1)-labeling with max{f(v):visinV(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 lambda-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.