ON RADIO NUMBER OF POWER OF CYCLES

September 26, 2011 10:52 WSPC/INSTRUCTION FILE

14˙Sahpankum

Asian-European Journal of Mathematics
Vol. 4, No. 3 (2011) 523–544 c World Scientific Publishing Company
DOI: 10.1142/S1793557111000435

ON RADIO NUMBER OF POWER OF CYCLES

Laxman Saha∗ , Pratima Panigrahi† and Pawan Kumar‡
Department of Mathematics, IIT Kharagpur
Kharagpur, West Bengal 721302, India
∗[e-mail]
†[e-mail]
‡[e-mail]

Communicated by M. R. R. Moghaddam
Received September 25, 2010
Revised June 1, 2011

A number of graph coloring problems have their roots in a communication problem known as the channel assignment problem. The channel assignment problem is the problem of assigning channels (non-negative integers) to the stations in an optimal way such that interference is avoided, see Hale [4]. The radio coloring of a graph is a special type of channel assignment problem. Here we develop a technique to find an upper bound for radio number of an arbitrary graph and also we give a lower bound for the same. r Applying these bounds we have obtained radio number of Cn , r 3, for several values r of n and r. Moreover for diameter 2 or 3 radio number of Cn have been determined completely for all values of n and r.
Keywords: Channel assignment; radio coloring; span; radio number.
AMS Subject Classification: MSC 2000: 05C78, 05C12, 05C15

1. Introduction
Radio coloring of a graph is a variation of the channel assignment problem. Let G be a simple connected graph with diameter D. A radio coloring of G is an assignment f of distinct non-negative integers to the vertices of G, such that for every two distinct vertices u and v of G
|f (u) − f (v)|

1 + D − d(u, v)

where d(u, v) is the distance between u and v in G (recall that distance between a pair of vertices in a graph is the length of a shortest path between them ). The span of a radio coloring f , span(f ), is the maximum integer assigned to a vertex of

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