Copyright (c) by Daphne Liu and Melanie Xie Radio Numbers for Square Paths & Cycles Daphne Liu & Melanie Xie California State University, Los Angeles Department of Mathematics This project is supported by the National Science Foundation (NSF) under grant DMS Motivation & Definitions Previous Known Results New Results References

Channel Assignment Problem? Motivation: This vertex coloring problem is motivated by the Channel Assignment Problem which is introduced by Hale in In this problem, several radio stations (or transmitters) are given. We want to assign to each station a non-negative integers as channels so that the interference is avoided. The closer the two station are, the stronger the interference. Hence, the closer the two stations, the larger the separation of channels. Goal: 1.To assign channels for a set of radio stations to prevent the interference between radio channels. 2.To minimized the span of channels.

A graph G=(V,E) consists of two sets, V and E, called vertex-set and edge-set, respectively. Elements of V are called vertices. Elements of E are called edges. The distance a graph G, denoted by d(u,v), between two vertices u and v in is the length of a shortest path from u to v. The diameter a graph G, denoted by diam(G), is the maximum distance between a pair of vertices. diam(C6)= 3 uv w xy z d(u,w) = 2 Definitions

Radio Labeling Given G=(V, E), a radio k-labeling is a function so that for any two vertices u and v, Example: diam(P 5 )=4

Radio Number For a simple graph G, the radio number of G, denoted by rn(G), is the minimum k of a radio k-labeling of G. Example: rn(G) = 7 (Chartrand et al. [2002]) diam(G) = 3